  We study surjective homomorphisms f:\prod_I A_i\to B of
not-necessarily-associative algebras over a commutative ring k, for I a
generally infinite set; especially when k is a field and B is
countable-dimensional over k.
  Our results have the following consequences when k is an infinite field, the
algebras are Lie algebras, and B is finite-dimensional:
  If all the Lie algebras A_i are solvable, then so is B.
  If all the Lie algebras A_i are nilpotent, then so is B.
  If k is not of characteristic 2 or 3, and all the Lie algebras A_i are
finite-dimensional and are direct products of simple algebras, then (i) so is
B, (ii) f splits, and (iii) under a weak cardinality bound on I, f is
continuous in the pro-discrete topology. A key fact used in getting (i)-(iii)
is that over any such field, every finite-dimensional simple Lie algebra L can
be written L=[x_1,L]+[x_2,L] for some x_1, x_2\in L, which we prove from a
recent result of J.M.Bois.
  The general technique of the paper involves studying conditions under which a
homomorphism on \prod_I A_i must factor through the direct product of finitely
many ultraproducts of the A_i.
  Several examples are given, and open questions noted.
