  Let $G$ be a non-compact classical semisimple Lie group and let $G/V$ be the
adjoint orbit with respect to a fixed element in $G$. These manifolds can be
equipped with an almost-K\"ahler structure and we provide explicit formulae for
the existence of special almost-complex structures on $G/V$ purely in terms of
the combinatorics of the associated Vogan diagram. The formulae are given
separately for Lie groups whose Lie algebras are of type $A_{\ell}$,
$B_{\ell}$, $C_{\ell}$, $D_{\ell}$, where $\ell$ denotes the rank of the Lie
algebra.
