Centralizers in reductive Liegroups = unimodular? Let $G$ be a real reductive group. Why is the centralizer of an element unimodular? What is a reference?

As the OP pointed out, my original answer was wrong.  In fact centralizers of elements are unimodular, and a reference can be found by following the link in the OP's comment below.  
Here is an example that I find interesting:
Consider the element $\begin{pmatrix} 1 & 0 & 1 \\0 & 1 & 0 \\0 & 0 & 1 \end{pmatrix}$ in $GL_3(\mathbb R)$.  Its centralizer is $\Big\{ \begin{pmatrix}
a & b & c \\ 0 & d & e \\ 0 & 0 & a \end{pmatrix} \Big\} \subset GL_3(\mathbb R)$.
Although this is a solvable group, and looks very similar to the Borel in $GL_3$, which is not unimodular, it is in fact a unimodular group (as the OP points out in a second comment below).
