  There are two Rellich inequalities for the bilaplacian, that is for $\int
(\Delta u)^2dx$, the one involving $|\nabla u|$ and the other involving $|u|$
at the RHS. In this article we consider these inequalities with sharp constants
and obtain sharp Sobolev-type improvements. More precisely, in our first result
we improve the Rellich inequality with $|\nabla u|$ obtained recently by Cazacu
in dimensions $n=3,4$ by a sharp Sobolev term thus complementing existing
results for the case $n\geq 5$. In the second theorem the sharp constant of the
Sobolev improvement for the Rellich inequality with $|u|$ is obtained.
