Centralizer in a Lie algebra

Let be a Lie algebra over and be a vector subspace. The centralizer of in is then the Lie subalgebra of elements giving zero bracket with all elements of , #m/def/lie i.e.

Proof of Lie subalgebra

Let . By the Jacobi identity

whence .

A related notion is the Centre of a Lie algebra , which includes only those elements that give zero bracket for all elements. A weakening is the Normalizer in a Lie algebra.

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