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.
Introduction. LetA be an associative algebra with involution * over a field F . The set L(A, * ) = {a ∈ A | a * = −a} of skew-symmetric elements with respect to * is clearly closed under the operation [a, b] = ab − ba, and L(A, * ) is a subalgebra in A (−) . The algebra L(A, * ) with respect to the operation [a, b] will be called the Lie algebra of skew-symmetric elements of A.In what follows all algebras are considered over a field F of characteristic 0. The standard notation and definitions are taken from [1]. We use the left-normed bracketing in nonassociative words.Denote by SJ[X n ] and Ass[X n ] the free special Jordan algebra and the free associative algebra on the set of generators X n = {x 1 , x 2 , . . . , x n } respectively. Let H[X n ] = H(Ass[X n ], * ) and L[X n ] = L(Ass[X n ], * ) be the Jordan algebra of symmetric elements and the Lie algebra of skew-symmetric elements of Ass[X n ] with respect to the standard involution * . Given an F -algebra A and B ⊆ A, we denote by (B) F the F -module generated by B. In this article we construct a simple system of generators of the F -module L = L(Ass[X 2 ], * ). A simple system of generators is known in the case H[X n ] with n ≤ 3. By the Shirshov-Cohn theorem [1, 2] H[X n ] = SJ[X n ] where n ≤ 3, i.e., H[X n ] = (a; a ∈ SJ[X n ]) F and SJ[X 4 ] H[X 4 ]. In the case L[X n ] no simple system of generators of the F -module L[X n ] was known. In the present article we prove that the set of elements [a, b], [a, b] 3 , where a, b ∈ SJ[X 2 ], generates the F -module L, i.e., L = ([a, b], [a, b] 3 ; a, b ∈ SJ[X 2 ]) F .