By a complex reductive algebraic group I mean the group of complex points of a (possibly disconnected) affine algebraic group defined over $\mathbb{C}$ whose unipotent radical (maximal connected unipotent normal subgroup) is trivial.
I can't seem to find a clear source for the following fact that I believe to be true:
A complex algebraic group is reductive if and only if it is the complexification of a compact Lie group.