The maximum symmetry metric on real projective space $ \mathbb{RP}^n $ is the round metric.
The maximum symmetry metric on complex projective space $ \mathbb{CP}^n $ is the Fubini-Study metric.
https://mathoverflow.net/questions/433847/maximum-symmetry-metric-on-mathbbcpn
Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. See for example
Two metrics are considered to be equivalent if they are isometric up to a constant multiple.
I'm interested in manifolds $ M $ for which there is a unique up to equivalence metric with isometry group of dimension $ N(M) $.
Is the pushforward of the biinvariant metric of $ F_4 $ onto the Cayley projective plane $$ \mathbb{OP}^2 \cong F_4/Spin(9) $$ a maximum symmetry metric in this sense?
