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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Jacobi identity</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 16 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-16" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">16 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Identitat_de_Jacobi" title="Identitat de Jacobi – Catalan" lang="ca" hreflang="ca" data-title="Identitat de Jacobi" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Jacobi-Identit%C3%A4t" title="Jacobi-Identität – German" lang="de" hreflang="de" data-title="Jacobi-Identität" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Identidad_de_Jacobi" title="Identidad de Jacobi – Spanish" lang="es" hreflang="es" data-title="Identidad de Jacobi" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Relation_de_Jacobi" title="Relation de Jacobi – French" lang="fr" hreflang="fr" data-title="Relation de Jacobi" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%BC%EC%BD%94%EB%B9%84_%ED%95%AD%EB%93%B1%EC%8B%9D" title="야코비 항등식 – Korean" lang="ko" hreflang="ko" data-title="야코비 항등식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Identitas_Jacobi" title="Identitas Jacobi – Indonesian" lang="id" hreflang="id" data-title="Identitas Jacobi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Identit%C3%A0_di_Jacobi" title="Identità di Jacobi – Italian" lang="it" hreflang="it" data-title="Identità di Jacobi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Jacobi-identiteit" title="Jacobi-identiteit – Dutch" lang="nl" hreflang="nl" data-title="Jacobi-identiteit" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A4%E3%82%B3%E3%83%93%E6%81%92%E7%AD%89%E5%BC%8F" title="ヤコビ恒等式 – Japanese" lang="ja" hreflang="ja" data-title="ヤコビ恒等式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Identidade_de_Jacobi" title="Identidade de Jacobi – Portuguese" lang="pt" hreflang="pt" data-title="Identidade de Jacobi" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%BE%D0%B6%D0%B4%D0%B5%D1%81%D1%82%D0%B2%D0%BE_%D0%AF%D0%BA%D0%BE%D0%B1%D0%B8" title="Тождество Якоби – Russian" lang="ru" hreflang="ru" data-title="Тождество Якоби" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Jacobijeva_enakost" title="Jacobijeva enakost – Slovenian" lang="sl" hreflang="sl" data-title="Jacobijeva enakost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Jacobi-identiteten" title="Jacobi-identiteten – Swedish" lang="sv" hreflang="sv" data-title="Jacobi-identiteten" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Jacobi_%C3%B6zde%C5%9Fli%C4%9Fi" title="Jacobi özdeşliği – Turkish" lang="tr" hreflang="tr" data-title="Jacobi özdeşliği" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%BE%D1%82%D0%BE%D0%B6%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D0%AF%D0%BA%D0%BE%D0%B1%D1%96" title="Тотожність Якобі – Ukrainian" lang="uk" hreflang="uk" data-title="Тотожність Якобі" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Property of some binary operations</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Jacobi identity</b> is a property of a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associative property</a>, any order of evaluation gives the same result (parentheses in a multiple product are not needed). The identity is named after the German mathematician <a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Carl Gustav Jacob Jacobi</a>. He derived the Jacobi identity for Poisson brackets in his 1862 paper on differential equations.<sup id="cite_ref-Poisson1809_1-0" class="reference"><a href="#cite_note-Poisson1809-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hawkins1991_2-0" class="reference"><a href="#cite_note-Hawkins1991-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Cross_product" title="Cross product">cross product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\times b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>×</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\times b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b420244850c1a22be4c326f91e146db8b037f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.068ex; height:2.176ex;" alt="{\displaystyle a\times b}"></span> and the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie bracket operation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> both satisfy the Jacobi identity. In <a href="/wiki/Analytical_mechanics" title="Analytical mechanics">analytical mechanics</a>, the Jacobi identity is satisfied by the <a href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson brackets</a>. In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, it is satisfied by operator <a href="/wiki/Commutator#Ring_theory" title="Commutator">commutators</a> on a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> and equivalently in the <a href="/wiki/Phase_space_formulation" class="mw-redirect" title="Phase space formulation">phase space formulation</a> of quantum mechanics by the <a href="/wiki/Moyal_bracket" title="Moyal bracket">Moyal bracket</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jacobi_identity&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></span> be two <a href="/wiki/Binary_operation" title="Binary operation">binary operations</a>, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> be the <a href="/wiki/Neutral_element" class="mw-redirect" title="Neutral element">neutral element</a> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span>. The <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="Jacobi_identity"></span><span class="vanchor-text">Jacobi identity</span></span></b> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\times (y\times z)\ +\ y\times (z\times x)\ +\ z\times (x\times y)\ =\ 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>×</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mi>y</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>×</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mi>z</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\times (y\times z)\ +\ y\times (z\times x)\ +\ z\times (x\times y)\ =\ 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c297956e402995708088d0abbe1c34a72da43f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.263ex; height:2.843ex;" alt="{\displaystyle x\times (y\times z)\ +\ y\times (z\times x)\ +\ z\times (x\times y)\ =\ 0.}"></span></dd></dl> <p>Notice the pattern in the variables on the left side of this identity. In each subsequent expression of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\times (b\times c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>×</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\times (b\times c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48bea774c3c60817f327f396d1e3c083c53bc78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.724ex; height:2.843ex;" alt="{\displaystyle a\times (b\times c)}"></span>, the variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> are permuted according to the cycle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto y\mapsto z\mapsto x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>y</mi> <mo stretchy="false">↦</mo> <mi>z</mi> <mo stretchy="false">↦</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto y\mapsto z\mapsto x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daec050c8e16626953a89857b939eb13bf4b381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.745ex; height:2.176ex;" alt="{\displaystyle x\mapsto y\mapsto z\mapsto x}"></span>. Alternatively, we may observe that the ordered triples <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y,z,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y,z,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56eb650ab735791b22f46a92fbf0eca1081feba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (y,z,x)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z,x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (z,x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c17b62988c89d83ac7d092a30e0e287ce00b3425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (z,x,y)}"></span>, are the <a href="/wiki/Permutation#Parity_of_a_permutation" title="Permutation">even permutations</a> of the ordered triple <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Commutator_bracket_form">Commutator bracket form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jacobi_identity&action=edit&section=2" title="Edit section: Commutator bracket form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The simplest informative example of a <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> is constructed from the (associative) ring of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> matrices, which may be thought of as infinitesimal motions of an <i>n</i>-dimensional vector space. The × operation is the <a href="/wiki/Commutator" title="Commutator">commutator</a>, which measures the failure of commutativity in matrix multiplication. Instead of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1613c1ff4b6fbfb6c80a8da83e90ad28f0ab3483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.594ex; height:2.176ex;" alt="{\displaystyle X\times Y}"></span>, the Lie bracket notation is used: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,Y]=XY-YX.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>X</mi> <mi>Y</mi> <mo>−</mo> <mi>Y</mi> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,Y]=XY-YX.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649cef3a00d4024de15414fd6a37627571c9609e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.173ex; height:2.843ex;" alt="{\displaystyle [X,Y]=XY-YX.}"></span></dd></dl> <p>In that notation, the Jacobi identity is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]\ =\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>Z</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>Z</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]\ =\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d77e8c02499d0bc3b2bdfe1a144ba07f6cc5048" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.37ex; height:2.843ex;" alt="{\displaystyle [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]\ =\ 0}"></span></dd></dl> <p>That is easily checked by computation. </p><p>More generally, if <b><span class="texhtml">A</span></b> is an associative algebra and <b><span class="texhtml mvar" style="font-style:italic;">V</span></b> is a subspace of <b><span class="texhtml">A</span></b> that is closed under the bracket operation: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,Y]=XY-YX}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>X</mi> <mi>Y</mi> <mo>−</mo> <mi>Y</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,Y]=XY-YX}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/838f73010b4f791eeaf245317fb4b6e07c45d741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.526ex; height:2.843ex;" alt="{\displaystyle [X,Y]=XY-YX}"></span> belongs to <b><span class="texhtml mvar" style="font-style:italic;">V</span></b> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,Y\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296fd06438d09cae93b3c2b879d5304b5fe7940e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.415ex; height:2.509ex;" alt="{\displaystyle X,Y\in V}"></span>, the Jacobi identity continues to hold on <b><span class="texhtml mvar" style="font-style:italic;">V</span></b>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Thus, if a binary operation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,Y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,Y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94470b44d283fde62130212956058ca6b727da37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.081ex; height:2.843ex;" alt="{\displaystyle [X,Y]}"></span> satisfies the Jacobi identity, it may be said that it behaves as if it were given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle XY-YX}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>Y</mi> <mo>−</mo> <mi>Y</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XY-YX}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d77f289bd8ea08d8990087aaaad8720e391c38d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.347ex; height:2.343ex;" alt="{\displaystyle XY-YX}"></span> in some associative algebra even if it is not actually defined that way. </p><p>Using the <a href="/wiki/Anticommutativity" class="mw-redirect" title="Anticommutativity">antisymmetry property</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,Y]=-[Y,X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,Y]=-[Y,X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c433e248c0c7e659ef3f2d9df64d8c7505630bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.068ex; height:2.843ex;" alt="{\displaystyle [X,Y]=-[Y,X]}"></span>, the Jacobi identity may be rewritten as a modification of the <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associative property</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [[X,Y],Z]=[X,[Y,Z]]-[Y,[X,Z]]~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>−</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [[X,Y],Z]=[X,[Y,Z]]-[Y,[X,Z]]~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe3847dca1561d17f643d4fd36a7851ff9c125d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.433ex; height:2.843ex;" alt="{\displaystyle [[X,Y],Z]=[X,[Y,Z]]-[Y,[X,Z]]~.}"></span></dd></dl> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,Z]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,Z]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366ab83f1ee0a471efb306249972deec6a5aad0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.988ex; height:2.843ex;" alt="{\displaystyle [X,Z]}"></span> is the action of the infinitesimal motion <b><span class="texhtml mvar" style="font-style:italic;">X</span></b> on <b><span class="texhtml mvar" style="font-style:italic;">Z</span></b>, that can be stated as: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p> The action of <i>Y</i> followed by <i>X</i> (operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,[Y,\cdot \ ]]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>,</mo> <mo>⋅</mo> <mtext> </mtext> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,[Y,\cdot \ ]]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0e9f50bf9ad3b7be2739a84cc0bf21337175fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.636ex; height:2.843ex;" alt="{\displaystyle [X,[Y,\cdot \ ]]}"></span>), minus the action of <i>X</i> followed by <i>Y</i> (operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ([Y,[X,\cdot \ ]]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mo>⋅</mo> <mtext> </mtext> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ([Y,[X,\cdot \ ]]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecebe6410bec49b9ed154b3d342a98ee11b73131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.541ex; height:2.843ex;" alt="{\displaystyle ([Y,[X,\cdot \ ]]}"></span>), is equal to the action of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,Y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,Y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94470b44d283fde62130212956058ca6b727da37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.081ex; height:2.843ex;" alt="{\displaystyle [X,Y]}"></span>, (operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [[X,Y],\cdot \ ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mo>⋅</mo> <mtext> </mtext> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [[X,Y],\cdot \ ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40034a989220666e8f7cb0082d27133dad27cf6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.636ex; height:2.843ex;" alt="{\displaystyle [[X,Y],\cdot \ ]}"></span>). </p></blockquote> <p>There is also a plethora of <a href="/wiki/Lie_superalgebra#properties" title="Lie superalgebra">graded Jacobi identities</a> involving <a href="/wiki/Anticommutator" class="mw-redirect" title="Anticommutator">anticommutators</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X,Y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X,Y\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d8641d8086abfedadc49fc18a63419662acb184" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.112ex; height:2.843ex;" alt="{\displaystyle \{X,Y\}}"></span>, such as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\{X,Y\},Z]+[\{Y,Z\},X]+[\{Z,X\},Y]=0,\qquad [\{X,Y\},Z]+\{[Z,Y],X\}+\{[Z,X],Y\}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mo fence="false" stretchy="false">{</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mo fence="false" stretchy="false">{</mo> <mi>Z</mi> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="2em" /> <mo stretchy="false">[</mo> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">[</mo> <mi>Z</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> <mo>+</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">[</mo> <mi>Z</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>Y</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\{X,Y\},Z]+[\{Y,Z\},X]+[\{Z,X\},Y]=0,\qquad [\{X,Y\},Z]+\{[Z,Y],X\}+\{[Z,X],Y\}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/815a6cdb80c32e53aeecd2ccccd07f4cb43dbeff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:92.93ex; height:2.843ex;" alt="{\displaystyle [\{X,Y\},Z]+[\{Y,Z\},X]+[\{Z,X\},Y]=0,\qquad [\{X,Y\},Z]+\{[Z,Y],X\}+\{[Z,X],Y\}=0.}"></span></dd></dl> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket of vector fields</a> and <a href="/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula" title="Baker–Campbell–Hausdorff formula">Baker–Campbell–Hausdorff formula</a></div> <div class="mw-heading mw-heading2"><h2 id="Adjoint_form">Adjoint form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jacobi_identity&action=edit&section=3" title="Edit section: Adjoint form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most common examples of the Jacobi identity come from the bracket multiplication <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7bd6292c6023626c6358bfd3943a031b27d663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.813ex; height:2.843ex;" alt="{\displaystyle [x,y]}"></span> on <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a> and <a href="/wiki/Lie_ring" class="mw-redirect" title="Lie ring">Lie rings</a>. The Jacobi identity is written as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>z</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>z</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a425004c6877ee5ecace799ca0396a3db1e4155b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.274ex; height:2.843ex;" alt="{\displaystyle [x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0.}"></span></dd></dl> <p>Because the bracket multiplication is <a href="/wiki/Anticommutativity" class="mw-redirect" title="Anticommutativity">antisymmetric</a>, the Jacobi identity admits two equivalent reformulations. Defining the <a href="/wiki/Adjoint_representation_of_a_Lie_algebra" class="mw-redirect" title="Adjoint representation of a Lie algebra">adjoint operator</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} _{x}:y\mapsto [x,y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ad</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>:</mo> <mi>y</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{x}:y\mapsto [x,y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9814bb34e60a706242e0a589bd22cbd90528f6fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.147ex; height:2.843ex;" alt="{\displaystyle \operatorname {ad} _{x}:y\mapsto [x,y]}"></span>, the identity becomes: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} _{x}[y,z]=[\operatorname {ad} _{x}y,z]+[y,\operatorname {ad} _{x}z].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ad</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>ad</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo>,</mo> <msub> <mi>ad</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{x}[y,z]=[\operatorname {ad} _{x}y,z]+[y,\operatorname {ad} _{x}z].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68a463244ff920979eda9e6ce2d8f64c399d27e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.956ex; height:2.843ex;" alt="{\displaystyle \operatorname {ad} _{x}[y,z]=[\operatorname {ad} _{x}y,z]+[y,\operatorname {ad} _{x}z].}"></span></dd></dl> <p>Thus, the Jacobi identity for Lie algebras states that the action of any element on the algebra is a <a href="/wiki/Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivation</a>. That form of the Jacobi identity is also used to define the notion of <a href="/wiki/Leibniz_algebra" title="Leibniz algebra">Leibniz algebra</a>. </p><p>Another rearrangement shows that the Jacobi identity is equivalent to the following identity between the operators of the adjoint representation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} _{[x,y]}=[\operatorname {ad} _{x},\operatorname {ad} _{y}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ad</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>ad</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ad</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{[x,y]}=[\operatorname {ad} _{x},\operatorname {ad} _{y}].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41644db0d06bdb5555514ac8bf9e5f2bcd81dc3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.021ex; height:3.176ex;" alt="{\displaystyle \operatorname {ad} _{[x,y]}=[\operatorname {ad} _{x},\operatorname {ad} _{y}].}"></span></dd></dl> <p>There, the bracket on the left side is the operation of the original algebra, the bracket on the right is the commutator of the composition of operators, and the identity states that the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ad} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ad} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dfabb59497ffe0c094cac2720f0c6a67b33e205" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:2.176ex;" alt="{\displaystyle \mathrm {ad} }"></span> map sending each element to its adjoint action is a <a href="/wiki/Lie_algebra_homomorphism" class="mw-redirect" title="Lie algebra homomorphism">Lie algebra homomorphism</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Related_identities">Related identities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jacobi_identity&action=edit&section=4" title="Edit section: Related identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Commutator#Identities_(group_theory)" title="Commutator">Hall–Witt identity</a> is the analogous identity for the <a href="/wiki/Commutator" title="Commutator">commutator</a> operation in a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>.</li></ul> <ul><li>The following identity follows from anticommutativity and Jacobi identity and holds in arbitrary Lie algebra:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,[y,[z,w]]]+[y,[x,[w,z]]]+[z,[w,[x,y]]]+[w,[z,[y,x]]]=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>z</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>w</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>w</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>z</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,[y,[z,w]]]+[y,[x,[w,z]]]+[z,[w,[x,y]]]+[w,[z,[y,x]]]=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf0522ac2ca9742f7c7406e222efde94e993534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.31ex; height:2.843ex;" alt="{\displaystyle [x,[y,[z,w]]]+[y,[x,[w,z]]]+[z,[w,[x,y]]]+[w,[z,[y,x]]]=0.}"></span></dd></dl> <ul><li>The Jacobi identity is equivalent to the <a href="/wiki/Product_Rule" class="mw-redirect" title="Product Rule">Product Rule</a>, with the Lie bracket acting as both a product and a derivative: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,[Y,Z]]=[[X,Y],Z]+[Y,[X,Z]]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,[Y,Z]]=[[X,Y],Z]+[Y,[X,Z]]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e48e2b35e36d68f9103e88c6dc4b49cf80cd5ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.206ex; height:2.843ex;" alt="{\displaystyle [X,[Y,Z]]=[[X,Y],Z]+[Y,[X,Z]]}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8705438171d938b7f59cd1bfa5b7d99b6afa5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}"></span> are vector fields, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,Y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,Y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94470b44d283fde62130212956058ca6b727da37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.081ex; height:2.843ex;" alt="{\displaystyle [X,Y]}"></span> is literally a derivative operator acting on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>, namely the <a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{X}Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{X}Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd1e6c305b263c2c55fdac7e57f69fbc6a505c4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.009ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{X}Y}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jacobi_identity&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Structure_constants" title="Structure constants">Structure constants</a></li> <li><a href="/wiki/Super_Jacobi_identity" class="mw-redirect" title="Super Jacobi identity">Super Jacobi identity</a></li> <li><a href="/wiki/Three_subgroups_lemma" title="Three subgroups lemma">Three subgroups lemma</a> (Hall–Witt identity)</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jacobi_identity&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Poisson1809-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Poisson1809_1-0">^</a></b></span> <span class="reference-text"><a href="#jacobi1862">C. G. J. Jacobi (1862), §26, Theorem V.</a></span> </li> <li id="cite_note-Hawkins1991-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hawkins1991_2-0">^</a></b></span> <span class="reference-text"><a href="#hawkins1991">T. Hawkins (1991)</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a> Example 3.3</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAlekseevIvanov2016" class="citation arxiv cs1">Alekseev, Ilya; Ivanov, Sergei O. (18 April 2016). "Higher Jacobi Identities". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1604.05281">1604.05281</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.GR">math.GR</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Higher+Jacobi+Identities&rft.date=2016-04-18&rft_id=info%3Aarxiv%2F1604.05281&rft.aulast=Alekseev&rft.aufirst=Ilya&rft.au=Ivanov%2C+Sergei+O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJacobi+identity" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2015" class="citation cs2">Hall, Brian C. (2015), <i>Lie Groups, Lie Algebras, and Representations: An Elementary Introduction</i>, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3319134666" title="Special:BookSources/978-3319134666"><bdi>978-3319134666</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+Groups%2C+Lie+Algebras%2C+and+Representations%3A+An+Elementary+Introduction&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer&rft.date=2015&rft.isbn=978-3319134666&rft.aulast=Hall&rft.aufirst=Brian+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJacobi+identity" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="jacobi1862" class="citation journal cs1"><a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi, C. G. J.</a> (1862). <a rel="nofollow" class="external text" href="https://eudml.org/doc/147847">"Nova methodus, aequationes differentiales partiales primi ordinis inter numerum variabilium quemcunque propositas integrandi"</a>. <i>Journal für die reine und angewandte Mathematik</i>. <b>60</b>: 1-181.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.atitle=Nova+methodus%2C+aequationes+differentiales+partiales+primi+ordinis+inter+numerum+variabilium+quemcunque+propositas+integrandi&rft.volume=60&rft.pages=1-181&rft.date=1862&rft.aulast=Jacobi&rft.aufirst=C.+G.+J.&rft_id=https%3A%2F%2Feudml.org%2Fdoc%2F147847&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJacobi+identity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="hawkins1991" class="citation journal cs1"><a href="/wiki/Thomas_W._Hawkins_Jr." title="Thomas W. Hawkins Jr.">Hawkins, Thomas</a> (1991). "Jacobi and the Birth of Lie's Theory of Groups". <i>Arch. Hist. Exact Sci</i>. <b>42</b> (3): 187-278. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00375135">10.1007/BF00375135</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Arch.+Hist.+Exact+Sci.&rft.atitle=Jacobi+and+the+Birth+of+Lie%27s+Theory+of+Groups&rft.volume=42&rft.issue=3&rft.pages=187-278&rft.date=1991&rft_id=info%3Adoi%2F10.1007%2FBF00375135&rft.aulast=Hawkins&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJacobi+identity" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jacobi_identity&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Jacobi_Identities"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/JacobiIdentities.html">"Jacobi Identities"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Jacobi+Identities&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FJacobiIdentities.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJacobi+identity" class="Z3988"></span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline 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