  Let K, K' be convex cones residing in finite-dimensional real vector spaces
E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable
if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K
and x_l' \in K' for all l. Let S(n), H(n), Q(n) be the spaces of n x n real
symmetric, complex hermitian and quaternionic hermitian matrices, respectively.
Let further S_+(n), H_+(n), Q_+(n) be the cones of positive semidefinite
matrices in these spaces. If a matrix in H(mn) = H(m) \otimes H(n) is H_+(m)
\otimes H_+(n)-separable, then it fulfills also the so-called PPT condition,
i.e. it is positive semidefinite and has a positive semidefinite partial
transpose. The same implication holds for matrices in the spaces S(m) \otimes
S(n), H(m) \otimes S(n), and for m < 3 in the space Q(m) \otimes S(n). We
provide a complete enumeration of all pairs (n,m) when the inverse implication
is also true for each of the above spaces, i.e. the PPT condition is sufficient
for separability. We also show that a matrix in Q(n) \otimes S(2) is Q_+(n)
\otimes S_+(2)-separable if and only if it is positive semidefinite.
