Appendix I Lie triple systems in compact semisimple Lie algebras The elementary theory of Lie triple systems is not easily found in the literature, and for this reason we present a brief self contained treatment of some parts of the theory that are relevant for this paper. Let G be a finite dimensional Lie algebra over Â such that the Killing form B is G negative definite. It is known that if G is any connected Lie group with Lie algebra G, then G is compact (See for example Proposition 6.6 and Theorem 6.9 of ([H, pp. 132-133]). A subspace W of G is called a Lie triple system in G if [X, [Y, Z]] ∈ W for all X,Y, Z ∈ W. Fix a bi-invariant metric on any compact connected Lie group G with Lie algebra G. If X = exp(W), where exp denotes the matrix exponential map, then it is a well known fact from the theory of Riemannian symmetric spaces that X is a totally geodesic submanifold of G ⇔ W is a Lie triple system in G. We are interested in the case that G = so(m), the Lie algebra of m x m skew symmetric matrices, and G is the special orthogonal group SO(m) for m ≥ 3. Our motivation for studying Lie triple systems here is somewhat different. We have already seen that if Cl V is the Clifford algebra determined by a finite dimensional real inner product space V and if j: Cl V → End (U) is a representation on a finite dimensional real vector space U, then there exists an inner product <,> on V such that W = j(Cl V) is a Lie triple system in O(U,<,>). Moreover, if m = dim V ≥ 2, then W + [W,W] = so(m+1) if m ≠ 3 and either W + [W,W] = so(4) or W = so(3) if m = 3. See Appendix II, Clifford algebras and Lie triple syhstems, for further details. The facts just stated are generalized by the following result, which is relevant for us in the case that G = so(m).

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