COMMUTATORS OF SKEW-INVOLUTIONS

Let SLn(F) be the group of all n x n matrices over a field F with determinant 1. Denote by I (I-n) the (n x n) identity matrix. A matrix A is called skew-involution if A(2) = -I. It is proved that every matrix in SL2n(F) is a product of at most three commutators of skew-involutions if F not equal Z(3) and SL2n(F)not equal SL2(Z(2)), and at most four commutators of skewinvolutions if F = Z(3) and n > 1. Every complex symplectic matrix is a product of two commutators of complex symplectic skew-involutions, and every real symplectic matrix is a product of not more than four commutators of real symplectic skewinvolutions.