It is well known fact that the finite dimensional Lie algebra will always be closed under commutation relation. But I have doubt about infinite Lie algebra. In some of the case it is closed under commutation relation and in other cases it not. Can we claim in general that infinite dimensional Lie algebra would not be closed under commutation ??
2026-05-04 21:23:21.1777929801
Infinite Lie algebra
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By definition, a vector space is a Lie algebra, if the Lie bracket is "closed under commutation relation", i.e., $[x,y]\in L$ for all $x,y\in L$, and satisfies skew-symmetry and the Jacobi identity. This holds for finite-dimensional and infinite-dimensional vector spaces.