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*](http://www.jmilne.org/math/CourseNotes/LAG.pdf).