  Let $\mathfrak{g}$ be a compact, simple Lie algebra of dimension $d$. It is a
classical result that the convolution of any $d$ non-trivial, $G$ -invariant,
orbital measures is absolutely continuous with respect to Lebesgue measure on
$\mathfrak{g}$ and the sum of any $d$ non-trivial orbits has non-empty
interior. The number $d$ was later reduced to the rank of the Lie algebra (or
rank $+1$ in the case of type $A_{n}$). More recently, the minimal integer
$k=k(X)$ such that the $k$-fold convolution of the orbital measure supported on
the orbit generated by $X$ is an absolutely continuous measure was calculated
for each $X\in \mathfrak{g}$.
  In this paper $\mathfrak{g}$ is any of the classical, compact, simple Lie
algebras. We characterize the tuples $(X_{1},...,X_{L})$, with $X_{i}\in
\mathfrak{g},$ which have the property that the convolution of the $L$ -orbital
measures supported on the orbits generated by the $X_{i}$ is absolutely
continuous and, equivalently, the sum of their orbits has non-empty interior.
The characterization depends on the Lie type of $ \mathfrak{g}$ and the
structure of the annihilating roots of the $X_{i}$. Such a characterization was
previously known only for type $A_{n}$.
