A Lie group $G$ is called reductive if its Lie algebra $\mathfrak{g}$ is reductive, i.e., a direct sum of an abelian and a semisimple Lie algebra.
A Lie algebra is reductive if and only if its adjoint representation is completely reducible, but this does not imply that all of its finite dimensional representations are completely reducible.
The concept of reductive is not quite the same for Lie groups as it is for algebraic groups (see at reductive algebraic group) because a reductive Lie group can be the group of real points of a unipotent algebraic group.
Review includes
Discussion of the “basic” multiplicative holomorphic line 2-bundle (Chern-Simons line 3-bundle) on complex reductive groups is in
Jean-Luc Brylinski, around theorem 5.4.10 (p. 226-227) of Loop spaces and characteristic classes, Birkhäuser
Jean-Luc Brylinski, Gerbes on complex reductive Lie groups (arXiv:math/0002158)