Nilpotency of some Lie algebras associated with p-groups

Let L = L-0 + L-1 be a Z(2)-graded Lie algebra over a commutative ring with unity in which 2 is invertible. Suppose that Lo is abelian and L is genera ted by finitely many homogeneous elements a(1),...,a(k) such that every com mutator in a(1),...,a(k) is ad-nilpotent. We prove that L is nilpotent. Thi s implies that any periodic residually finite 2'-group G admitting an invol utory automorphism rb with Cc(rp) abelian is locally finite.