  Let $G_\cpl$ be a connected complex reductive Lie group, and $G$ be a real
form. Let $(\pi,V)$ be a finite-dimensional irreducible representation of $G$.
Assume $(\pi,V)$ admits a $G$ invariant hermitian form. {In
\cite{Signature-of-a-rep-of-reductive}, an analog of the Weyl dimension formula
that instead computes the signature of the invariant form for finite
dimensional representations of complex Lie groups is given. The goal of this
paper is to give a short alternate proof as an application of a more general
twisted character theory of \cite{Dirac-Index-and-Twisted-Characters}.
