Linear Lie algebra
Any Lie algebra is a linear Lie algebra in the sense that there is always a faithful representation of (in fact, on a finite-dimensional vector space by Ado's theorem if is itself finite-dimensional.)
Let V be a finite-dimensional vector space over a field of characteristic zero and a subalgebra of . Then V is semisimple as a module over if and only if (i) it is a direct sum of the center and a semisimple ideal and (ii) the elements of the center are diagonalizable (over some extension field).
- Jacobson 1962, Ch III, Theorem 10