Subring of a Lie ring

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Let L be a Lie ring and S be a subset of L. We say that S is a subring of L, or a Lie subring, if it satisfies the following equivalent conditions:

  1. S is a subgroup of the additive group of L and is closed under the Lie bracket of L.
  2. S is a Lie ring with the addition and Lie bracket operations induced from L.