Are the dimensions of division algebras over the real numbers related to with generalizations of Euler's four square identity?

Question

I know that the only division algebras over the real numbers have dimension $1, 2, 4,$ and $8$ (real numbers, complex numbers, quaternions, octonions). I also know that those are the only numbers of squares for which a bilinear generalization of Euler's four square identity exists (product of squares, Brahmagupta-Fibonacci identity, Euler's four square identity, Degen's eight square identity). Are these two facts related in any way or is this just a coincidence? I feel that it might just be a coincidence because the only dimension for an algebra over algebraically closed fields is 1 and there are infinitely many dimensions for division algebras over fields which aren't algebraically closed or real closed.