We study Wick-rotations of left-invariant metrics on Lie groups, using results from real GIT (\cite{ 1}, \cite{ 2}, \cite{ 3}). An invariant for Wick-rotation of Lie groups is given, and we describe when a pseudo-Riemannian Lie group can be Wick-rotated to a Riemannian Lie group. We also prove a general version (for general Lie algebras) of $\acute{ E}$. Cartan's result, namely the existence and conjugacy of Cartan involutions.
Categories: math.DG