  The aim of this paper is to use the so-called Cayley transform to compute the
LS category of Lie groups and homogeneous spaces by giving explicit categorical
open coverings. When applied to U(n), $U(2n)/Sp(n)$ and $U(n)/O(n)$ this method
is simpler than those formerly known. We also show that the Cayley transform is
related to height functions in Lie groups, allowing to give a local linear
model of the set of critical points. As an application we give an explicit
covering of $Sp(2)$ by categorical open sets. The obstacles to generalize these
results to $Sp(n)$ are discussed.
