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In algebra, a linear Lie algebra is a subalgebra g {\displaystyle {\mathfrak {g}}} of the Lie algebra g l {\displaystyle {\mathfrak {gl}}} 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 g {\displaystyle {\mathfrak {g}}}
Let V be a finite-dimensional vector space over a field of characteristic zero and g {\displaystyle {\mathfrak {g}}} a subalgebra of g l {\displaystyle {\mathfrak {gl}}}. Then V is semisimple as a module over g {\displaystyle {\mathfrak {g}}} if and only if it is a direct sum of the center and a semisimple ideal and the elements of the center are diagonalizable.