Structurable algebra

In abstract algebra, a structurable algebra is a certain kind of unital involutive non-associative algebra over a field. For example, all Jordan algebras are structurable algebras (with the trivial involution), as is any alternative algebra with involution, or any central simple algebra with involution. An involution here means a linear anti-homomorphism whose square is the identity.

Assume A is a unital non-associative algebra over a field, and ${\displaystyle x\mapsto {\bar {x}}}$ is an involution. If we define ${\displaystyle V_{x,y}z:=(x{\bar {y}})z+(z{\bar {y}})x-(z{\bar {x}})y}$, and ${\displaystyle [x,y]=xy-yx}$, then we say A is a structurable algebra if:

${\displaystyle [V_{x,y},V_{z,w}]=V_{V_{x,y}z,w}-V_{z,V_{y,x}w}.}$