# Hermitian Octonion matrices?

The space of $$n\times n$$ octionion matrics $$A$$ such that $$AA^*=A^*A=I_n$$ is not a Lie group due to lack of associativity. But is it a smooth manifold? What is its dimension?

• Tangent space at the identity has antihermitian matrices, so dimension is $8\binom{n}{2}+7n$. I assume some annoying calculation shows it's a smooth variety. – arctic tern May 7 at 18:03
• Of course for $n = 1$ the set can be identified with $S^7$. – Travis May 7 at 18:11