Let $ L=L_0+L_1$ be a $mathbb{Z}_2$-graded Lie algebra over acommutative ring with unity in which $2$ is invertible. Supposethat $L_0$ is abelian and $L$ is generated by finitely manyhomogeneous elements $a_1,dots,a_k$ such that every commutator in$a_1,dots,a_k$ is ad-nilpotent. We prove that $L$ is nilpotent.This implies that any periodic residually finite $2'$-group $G$admitting an involutory automorphism $phi$ with $C_G(phi)$abelian is locally finite.
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机译：设$ L = L_0 + L_1 $为在$ 2 $可逆的具有单位的交换环上的$ mathbb {Z} _2 $阶Lie代数。假设$ L_0 $是阿贝尔方，而$ L $是由有限个同质元素$ a_1，dots，a_k $生成的，因此$ a_1，dots，a_k $中的每个换向子都是ad-幂等的。我们证明$ L $是幂等的，这意味着任何周期性残差有限的$ 2'$-group $ G $接受具有$ C_G（phi）$ abelian的不强制自同构$ phi $都是局部有限的。
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