  Horn's problem, i.e., the study of the eigenvalues of the sum $C=A+B$ of two
matrices, given the spectrum of $A$ and of $B$, is re-examined, comparing the
case of real symmetric, complex Hermitian and self-dual quaternionic $3\times
3$ matrices. In particular, what can be said on the probability distribution
function (PDF) of the eigenvalues of $C$ if $A$ and $B$ are independently and
uniformly distributed on their orbit under the action of, respectively, the
orthogonal, unitary and symplectic group? While the two latter cases (Hermitian
and quaternionic) may be studied by use of explicit formulae for the relevant
orbital integrals, the case of real symmetric matrices is much harder. It is
also quite intriguing, since numerical experiments reveal the occurrence of
singularities where the PDF of the eigenvalues diverges. Here we show that the
computation of the PDF of the symmetric functions of the eigenvalues for
traceless $3\times 3$ matrices may be carried out in terms of algebraic
functions;- roots of quartic polynomials;- and their integrals. The computation
is carried out in detail in a particular case, and reproduces the expected
singular patterns. The divergences are of logarithmic or inverse power type. We
also relate this PDF to the (rescaled) structure constants of zonal polynomials
and introduce a zonal analogue of the Weyl ${\rm SU}(n)$ characters.
