Communicated by Efim Zelmanov MSC: primary 17A60, 17B05 secondary 03C20, 03E55, 08B25, 17B20, 17B30 Keywords: Homomorphic images of infinite direct products of nonassociative algebras Simple, solvable, and nilpotent Lie algebras Maps that factor through ultraproducts Measurable cardinalsWe study surjective homomorphisms f : I A i → 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 countabledimensional 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 finitedimensional simple Lie algebra L can be written L = [x 1 , L] + [x 2 , L] for some x 1 , x 2 ∈ 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 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.

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Abstract. Let G be any analytic group, let T be a maximal toroid of the radical of G, and let S be a maximal semisimple analytic subgroup of G.If, and Z(L) is the center of L, we show that G has a faithful representation if and (0), and (ii) for some maximal semisimple subalgebra S of L, the simply connected analytic group with Lie algebra S has a faithful representation. So it would be of interest to find a similar criterion for a single analytic group G to have a faithful representation. Such a criterion is given in Theorem 2 below. As a consequence, we obtain Moskowitz' Theorem in Corollary 3. So our criterion in the solvable case says that G has a faithful representation if and only if [L, L] ∩ Z(L) ∩ L(T ) = (0) for some maximal toroid T of G where L = L(G); whereas the well-known criterion in the solvable case is that G has a faithful representation if and only if [G, G] is closed in G and simply connected [2, p. 220]. For the case of semisimple analytic groups, we refer the reader to [2, pp. 199-201].Our proof uses the notion of nuclei of analytic groups introduced by Hochschild and Mostow. A nucleus K of an analytic group G is a closed normal simply connected solvable analytic subgroup of G such that G/K is reductive. An analytic group G has a faithful representation if and only if G has a nucleus; if G has a nucleus K, then G = K · P (semi-direct) for every maximal reductive analytic subgroup P of G [3, Section 2]. Recall that an analytic group is reductive if it has a faithful representation and all its representations are semisimple.

Abstract. We extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed field K of characteristic 0. However, some modifications are needed in some extensions. So we introduce the pro-discrete topology on the Lie algebra L(G) of the pro-affine algebraic group G over K, which is discrete in the finite-dimensional case and linearly compact in general. As an example, if L is any sub Lie algebra of L(G), we show that the closure of

First we give a new proof of Goto's theorem for Lie algebras of compact semisimple Lie groups using Coxeter transformations. Namely, every x in L = Lie(G) can be written as x = [a, b] for some a, b in L. By using the same method, we give a new proof of the following theorem (thus avoiding the classification tables of fundamental weights): in compact semisimple Lie algebras, orthogonal Cartan subalgebras always exist (where one of them can be chosen arbitrarily). Some of the consequences of this theorem are the following. (i) If L = Lie(G) is such a Lie algebra and C is any Cartan subalgebra of L, then the G-orbit of C ⊥ is all of L. (ii) The consequence in part (i) answers a question by L. Florit and W. Ziller on fatness of certain principal bundles. It also shows that in our case, the commutator map L × L → L is open at (0, 0). (iii) given any regular element x of L, there exists a regular element y such that L = [x, L] + [y, L] and x, y are orthogonal. Then we generalize this result about compact semisimple Lie algebras to the class of non-Hermitian real semisimple Lie algebras having full rank.Finally, we survey some recent related results , and construct explicitly orthogonal Cartan subalgebras in su(n), sp(n), so(n).