Semisimple Lie Algebras and the Root Space Decomposition
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چکیده
That it is a Lie homomorphism is precisely the statement of the Jacobi identity. Another exact restatement of the Jacobi identity is contained in the fact that ad takes values in the subalgebra Der(g) of derivations. The notion of a Lie algebra is meant to be an abstraction of the additive commutators of associative algebras. While lie algebras are almost never associative as algebras, they have the very property that commutators of associative algebras satisfy by taking place in an associative settingthe Jacobi identity.
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