Wick-rotations of pseudo-Riemannian Lie groups

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.