Exercise 7, § 3

Let A be the algebra of Cayley octonions (Algebra, Chap. III, App., no. 3), with the basis $(e_i)_{0\leq i\leq 7}$ of loc. cit. Denote by V the subspace of pure octonions generated by $e_1, . . . , e_7$ and by E the subspace of V generated by $e_1, e_2, e_3, e_5, e_6, e_7$. Identify the subalgebra of A generated by $e_0, e_4$ with the field $\mathbf{C}$ of complex numbers, and denote by G the topological group of automorphisms of the unital algebra A.

a) Denote by Q the quadratic form on V induced by the Cayley norm, so that $(e_i)_{1\leq i\leq 7}$ is an orthonormal basis of V. Prove that the map $\sigma \rightarrow \sigma |V$ is an injective homomorphism from G to the group $\mathbf{S}\mathbf{O}$(Q), which is isomorphic to $\mathbf{S}\mathbf{O}(7,\mathbf{R})$.

b) Show that the multiplication of A gives E the structure of a $\mathbf{C}$-vector space of dimension 3, of which $\{e_1, e_2, e_3\}$ is a basis. Denote by $\Phi$ the hermitian form on E for which this basis is orthonormal. Let H be the fixer of $e_4$ in G. Show that the map $\sigma \rightarrow \sigma |E$ is an isomorphism from H to the group $\mathbf{S}\mathbf{U}(\Phi )$, which is isomorphic to $\mathbf{S}\mathbf{U}(3,\mathbf{C})$. The map $\sigma \rightarrow \sigma (e_4)$ induces an embedding of $G/H$ in the sphere of V, which is isomorphic to $\mathbf{S}_6$. c) Let T be the torus in H consisting of the automorphisms $\sigma$ such that $\sigma (e_0) =e_0,\sigma (e_1) =\alpha e_1,\sigma (e_2) =\beta e_2,\sigma (e_3) =\gamma e_3,\sigma (e_4) =e_4,\sigma (e_5) =$ $\overline{\alpha e}_5,\sigma (e_6) = \overline{\beta e}_6,\sigma (e_7) = \overline{\gamma e}_7$, where $\alpha , \beta , \gamma$ are three complex numbers of modulus 1 such that $\alpha \beta \gamma = 1$. Let N be the normalizer of T in G; show that $N/T$ is of order 12 (note that each element of N must stabilize the set of the $\pm e_i,i\not= 0$); deduce that G is of rank 2.

d) Show that G is connected semi-simple of type $G_2$ and that $G/H$ can be identified with $\mathbf{S}_6$ (show that $G_0\not= H$; deduce that $G_0$ is of type $G_2$, then use a dimension argument). Every compact group of type $G_2$ is isomorphic to G.