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A [[triple system]] considered for constructing all simple Lie algebras (cf. [[Lie algebra|Lie algebra]]), and introduced as an algebraic system which is a generalization both of the algebraic systems appearing in the metasymplectic geometry developed by H. Freudenthal and of a generalized [[Jordan triple system|Jordan triple system]] of second order developed by I.L. Kantor.

Recall that a triple system is a [[Vector space|vector space]] $V$ over a [[Field|field]] $\Phi$ together with a $\Phi$-trilinear mapping $V \times V \times V \rightarrow V$.

For $ \epsilon = \pm 1$, a vector space $U ( \varepsilon )$ over a field $\Phi$ with the trilinear product $\langle x y z \rangle$ is called a Freudenthal–Kantor triple system if

\begin{equation} \tag{a1} \langle a b \langle c d e \rangle \rangle = \langle \langle a b c \rangle \rangle + \varepsilon \langle c \langle b a d \rangle e \rangle + \langle c d \langle a b e \rangle \rangle, \end{equation}

\begin{equation} \tag{a2} K ( L ( a , b ) c , d ) + K ( c , L ( a , b ) d ) + K ( a , K ( c , d ) b ) = 0, \end{equation}

where $L ( a , b ) c = \langle a b c \rangle$ and $K ( a , b ) c = \langle a c b \rangle - \langle b c a \rangle$.

In particular, a Freudenthal–Kantor triple system $U ( \varepsilon )$ is said to be balanced if there exists a bilinear form $\langle \, .\, ,\,  . \, \rangle$ such that $K ( a , b ) = \langle a , b \rangle \operatorname{Id}$, for all $a,b \in U ( \varepsilon )$.

This balancing property is closely related to metasymplectic geometry.

Note that if $\varepsilon = - 1$ and $K ( a , b ) \equiv 0$ (identically), then the Freudenthal–Kantor triple system reduces to a Jordan triple system.

As the notion of a Freudenthal–Kantor triple system includes the notions of a generalized Jordan triple system of second order, a structurable algebra, and an [[Allison–Hein triple system|Allison–Hein triple system]], it is useful in obtaining all Lie algebras, without the use of root systems and Cartan matrices.

Let $V$ be a vector space with a bilinear form $\langle x , y \rangle = - \varepsilon \langle y , x \rangle$. Then $V$ is a Freudenthal–Kantor triple system with respect to the triple product $\langle x y z \rangle : = \langle y , z \rangle x$. In particular, it is important that the linear span $\mathbf{k} : = \{ K ( a , b ) \} _ { \text{span} }$ of the set $K ( a , b )$ makes a Jordan triple system of $( \text { End } U ( \varepsilon ) ) ^ { + }$ with respect to the triple product $\{ A B C \} : = 1 / 2 ( A B C + C B A )$.

Let $U ( \varepsilon )$ be a Freudenthal–Kantor triple system. The vector space $U ( \varepsilon ) \oplus U ( \varepsilon )$ becomes a [[Lie triple system|Lie triple system]] with respect to the triple product defined by

\begin{equation*} \left[ \left( \begin{array} { l } { a } \\ { b } \end{array} \right) \left( \begin{array} { l } { c } \\ { d } \end{array} \right) \left( \begin{array} { l } { e } \\ { f } \end{array} \right) \right] : = \end{equation*}

\begin{equation*} := \left( \begin{array} { c c } { L ( a , d ) - L ( c , b ) } &amp; { K ( a , c ) } \\ { - \varepsilon K ( b , d ) } &amp; { \varepsilon ( L ( d , a ) - L ( b , c ) ) } \end{array} \right) \left( \begin{array} { l } { e } \\ { f } \end{array} \right). \end{equation*}

Using this, one can obtain the Lie triple system $U ( \varepsilon ) \oplus U ( \varepsilon )$ associated with $U ( \varepsilon )$; it is denoted be $T ( \varepsilon )$.

Using the concept of the standard embedding Lie algebra $L ( \varepsilon ) = \operatorname { Inn } \operatorname { Der } T ( \varepsilon ) \oplus T ( \varepsilon )$ associated with a Lie triple system $T ( \varepsilon )$, one can obtain the construction of $L ( \varepsilon )$ associated with a Freudenthal–Kantor triple system $U ( \varepsilon )$. In fact, put

$L_{2}$ equal to the linear span of the endomorphisms

\begin{equation*} \left( \begin{array} { c c } { 0 } &amp; { K ( a , b ) } \\ { 0 } &amp; { 0 } \end{array} \right); \end{equation*}

$L _ { 1 } : = U ( \varepsilon ) \oplus ( 0 )$;

$L_{ - 1} : = ( 0 ) \oplus U ( \varepsilon )$;

$L_0$ equal to the linear span of the endomorphisms

\begin{equation*} \left( \begin{array} { c c } { L ( a , b ) } &amp; { 0 } \\ { 0 } &amp; { \varepsilon L ( b , a ) } \end{array} \right); \end{equation*}

$L_{ - 2}$ equal to the linear span of the endomorphisms

\begin{equation*} \left( \begin{array} { r r } { 0 } &amp; { 0 } \\ { - \varepsilon K ( c , d ) } &amp; { 0 } \end{array} \right). \end{equation*}

Then one obtains the decomposition

\begin{equation*} L ( \varepsilon ) = L _ { - 2 } \bigoplus L _ { - 1 } \bigoplus L _ { 0 } \bigoplus L _ { 1 } \bigoplus L _ { 2 }, \end{equation*}

and, more precisely,

\begin{equation*} \left[ \left( \begin{array} { c c } { \text{Id} } &amp; { 0 } \\ { 0 } &amp; { -  \text{Id} } \end{array} \right) , L _ { i } \right] = i L _ { i } ( - 2 \leq i \leq 2 ). \end{equation*}

These results imply the dimensional formula

\begin{equation*} \operatorname { dim } _ { \Phi } L ( \varepsilon ) = 2 ( \operatorname { dim } _ { \Phi } U ( \varepsilon ) + \operatorname { dim } _ { \Phi } \{ K ( x , y ) \} _ { \operatorname { span } } )+ \end{equation*}

\begin{equation*} + \operatorname { dim } _ { \Phi } \{ L ( x , y ) \} _ { \operatorname { span } } = \end{equation*}

\begin{equation*} = \operatorname { dim } _ { \Phi } T ( \varepsilon ) + \operatorname { dim } _ { \Phi } \operatorname { Inn } \operatorname { Der } T ( \varepsilon ). \end{equation*}

This algebra $L ( \varepsilon )$ is called the Lie algebra associated with $U ( \varepsilon )$.

The concepts of a triple system and a supertriple system are important in the theory of quarks and Yang–Baxter equations.

Note that a  "triple system"  in the sense discussed above is totally different from  "triple system"  in combinatorics (see, e.g., [[Steiner triple system]]).

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  H. Freudenthal,   "Beziehungen der $E _ { 7 }$ und $E _ { 8 }$ zur Oktavenebene I–II"  ''Indag. Math.'' , '''16'''  (1954)  pp. 218–230; 363–386</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  N. Kamiya,   "The construction of all simple Lie algebras over $C$ from balanced Freudenthal–Kantor triple systems" , ''Contributions to General Algebra'' , '''7''' , Hölder–Pichler–Tempsky, Wien  (1991)  pp. 205–213</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  N. Kamiya,   "On Freudenthal–Kantor triple systems and generalized structurable algebras" , ''Non-Associative Algebra and Its Applications'' , Kluwer Acad. Publ.  (1994)  pp. 198–203</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  N. Kamiya,   S. Okubo,   "On $\delta$-Lie supertriple systems associated with $( \varepsilon , \delta )$-Freudenthal–Kantor supertriple systems"  ''Proc. Edinburgh Math. Soc.'' , '''43'''  (2000)  pp. 243–260</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  I.L. Kantor,   "Models of exceptional Lie algebras"  ''Soviet Math. Dokl.'' , '''14'''  (1973)  pp. 254–258</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  S. Okubo,   "Introduction to octonion and other non-associative algebras in physics" , Cambridge Univ. Press  (1995)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  K. Yamaguti,   "On the metasymplectic geometry and triple systems"  ''Surikaisekikenkyusho Kokyuroku, Res. Inst. Math. Sci. Kyoto Univ.'' , '''306'''  (1977)  pp. 55–92  (In Japanese)</td></tr></table>