Using isometric group to describe E7.

Question

I read John C. Baez's paper, The Octonions, and I am wondering the following statement: $$E_7\simeq Isom(\mathbb{(H\otimes O)P}^2).$$ In his contents, I can only figure out $$E_7\hookrightarrow Isom(\mathbb{(H\otimes O)P}^2),$$ but the other side is unclear for me. So how to contribute the inclusion on the other side? Or are there any references to this question?

Answer 1

John C. Baez is giving the reference [6] for the proof in the section on $E_7$. Reference [6] is the book by Arthur L. Besse, Einstein Manifolds, Springer, Berlin, 1987, pp. $313-316$. Actually, the quateroctonionic projective plane here is defined by $$ (\mathbb{H }\otimes \mathbb{O})\mathbb{P}^2 = E_7 /((Spin(12) \times Sp(1))/\mathbb{Z}_2). $$