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?

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    $\begingroup$ 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. $\endgroup$ – arctic tern May 7 at 18:03
  • $\begingroup$ Of course for $n = 1$ the set can be identified with $S^7$. $\endgroup$ – Travis May 7 at 18:11

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