If a Lie subalgebra of $\mathfrak{gl}(V)$ is the Lie algebra of an algebraic group, then it contains the semisimple and nilpotent factors of any element.
There is a five-dimensional Lie algebra for which this fails, which you can find in Bourbaki Lie groups and algebras, I, 5, Exercise 6, or in 1.25 and 3.42 of my notes Milne Lie algebras, algeraic groups and Lie groups.
