# The Octonions by Michael Taylor

By Michael Taylor

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Now if h and h⊥ are both nonzero, Rank g = 2 =⇒ Rank h = Rank h⊥ = 1. But, as is well known, Rank h = 1 =⇒ dim h = 1 or 3, so we have the conclusion that dim G ≤ 6. 2 that the Lie algebra Der(O) of Aut(O) has no nontrivial ideals. A connected Lie group with this property is typically said to be simple. However, we can establish the more precise result that Aut(O) contains no nontrivial normal subgroups. Indeed, if H were such a subgroup, so would be its closure, so it suffices to consider the case when H is closed.

33 We return to the situation introduced three paragraphs above, with T ⊂ H ⊂ G, and π the restriction to H of the adjoint representation of G on g (and on its complexification gC ). 18) π = πh ⊕ π1 . Of course, πh is simply the adjoint action of H on h. We need to analyze π1 . To do this, it is convenient to look at the homogeneous space M = G/H, on which G acts transitively. Then H is the subgroup of elements of G that fix the point p = eH ∈ M . , a real representation ρ of H on Tp M . 19) π1 ≈ ρ.

Bryant, Submanifolds and special structures on the octonions, J. Diff. Geom. 17 (1982), 185–232. R. Bryant, On the geometry of almost-complex 6-manifolds, Asian J. Math. 10 (2006), 561–605. F. R. Harvey, Spinors and Calibrations, Academic Press, New York, 1990. H. B. Lawson and M. L. Michelson, Spin Geometry, Princeton Univ. Press, Princeton NJ, 1989. K. McCrimmon, Jordan algebras and their applications, Bull. AMS 84 (1978), 612–627. I. Porteous, Clifford Algebras and Classical Groups, Cambridge Univ.