The Lie algebra of skew-symmetric elements and its application in the theory of Jordan algebras

We prove that the Lie algebra of skew-symmetric elements of the free associative algebra of rank 2 with respect to the standard involution is generated as a module by the elements $[a,b]$ and $[a,b]^3$, where $a$ and $b$ are Jordan polynomials. Using this result we prove that the Lie algebra of Jordan derivations of the free Jordan algebra of rank 2 is generated as a characteristic $F$-module by two derivations. We show that the Jordan commutator $s$-identities follow from the Glennie–Shestakov $s$-identity.