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Theorem 16. Let be compact, be an elementary involutive pair, and let be the iso-involutive sum of an iso-involutive group of lower index 1 and of type 2. 10. 37) in general. However, we indicate only two important particular subcases (which almost solves the problem in the general case). 11. 6 (Theorem 3) (after some re-normalization of the base). Furthermore, where is elementary. This follows from the property that is an involutive pair of lower index 1. 6 (Theorem 3). 6 (Theorem 3), whence it follows that with a natural embedding.
Taking an arbitrary base in we induce a base in by the automorphism Furthermore, if are the identity involutive automorphisms then we take bases in and arbitrarily. 6 (Definition 19). 8 (Theorem 8) there exists the derived involutive group Using, furthermore, a Lie algebra and the derived involutive group we repeat the above construction etc.. ). Then since it follows that that is, they are orthogonal with respect to the Cartan metric of and so we obtain an orthonormal involutive base. 3. 4. 16 (Theorem 10).
14. Theorem 18. Let be a compact Lie algebra, be an elementary involutive pair, and let be an involutive sum of type 2 and of lower index 1, and Then with the natural embeddings. 1. Theorem 19. Let be a simple compact Lie algebra, and be an iso-involutive sum of lower index 1 and not of type 1 generated by an iso-involutive group Then and its corresponding iso-involutive sum is of lower index 1 and of type 1 (with the conjugating isomorphism Proof. Let us consider the sequence of automorphisms from such that under the action of by We denote the subspace of all elements of Then Evidently all these subspaces are invariant under the action of the following involutive pairs of the involutive automorphism immobile thus we have where Since and all these involutive pairs are of lower index 1.