Linear Lie algebra

In algebra, a linear Lie algebra is a subalgebra $$\mathfrak{g}$$ of the Lie algebra $$\mathfrak{gl}(V)$$ consisting of endomorphisms of a vector space V. In other words, a linear Lie algebra is the image of a Lie algebra representation.

Any Lie algebra is a linear Lie algebra in the sense that there is always a faithful representation of $$\mathfrak{g}$$ (in fact, on a finite-dimensional vector space by Ado's theorem if $$\mathfrak{g}$$ is itself finite-dimensional.)

Let V be a finite-dimensional vector space over a field of characteristic zero and $$\mathfrak{g}$$ a subalgebra of $$\mathfrak{gl}(V)$$. Then V is semisimple as a module over $$\mathfrak{g}$$ 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).