  Let $G$ be a real semisimple Lie group with trivial centre and no compact
factors. Given a conjugate pair of either real hyperbolic elements or unipotent
elements $a$ and $b$ in $G$ we find a conjugating element $g \in G$ such that
$d_G(1,g) \leq L(d_G(1,u)+d_G(1,v))$, where $L$ is a positive constant which
will depend on some property of $a$ and $b$. For the vast majority of such
elements however, $L$ can be assumed to be a uniform constant.
