Examples
The most basic example is the Lie algebra of matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an n-dimensional vector space, This is the Lie algebra of the general linear group GL(n), and is reductive as it decomposes as corresponding to traceless matrices and scalar matrices.
Any semisimple Lie algebra or abelian Lie algebra is a fortiori reductive.
Over the real numbers, compact Lie algebras are reductive.
Read more about this topic: Reductive Lie Algebra
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