Lie ring

Summary

A Lie ring is an abelian group with a biadditive bracket that is alternating and satisfies the Jacobi identity. It is a Lie algebra without a field of scalars, or equivalently a Lie algebra over the integers $\Z$. Every associative ring becomes a Lie ring under the commutator $xy - yx$. Lie rings appear in group theory, for example through the lower central series of a group.