A Characterization of Lie Algebras of Skew-Symmetric Elements

Abstract

A characterization of Lie algebras of skew-symmetric elements of associative algebras with involution is obtained. It is proved that a Lie algebra L is isomorphic to a Lie algebra of skew-symmetric elements of an associative algebra with involution if and only if L admits an additional (Jordan) trilinear operation {x,y,z} that satisfies the identities

$$\{x,y,z\}=\{z,y,x\},$$

$$[[x,y],z]=\{x,y,z\}-\{y,x,z\},$$

$$[\{x,y,z\},t]=\{[x,t],y,z\}+\{x,[y,t],z\}+\{x,y,[z,t]\},$$

$$\{\{x,y,z\},t,v\}=\{\{x,t,v\},y,z\}-\{x,\{y,v,t\},z\}+\{x,y,\{z,t,v\}\},$$

where [x,y] stands for the multiplication in L.