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Hi all.

I was wondering if anyone has seen any work related to either traceless matrices of Octonions (with trace defined as the sum of diagonal) or matrices of pure imaginary Octonions (meaning real part of the matrix is a matrix of zeros).

Thanks in advance.

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You can put octonions inside a matrix, but this might not be a particularly fruitful thing to do since the octonion algebra is not associative. (So in particular you don't get octonionic Lie algebras or Lie groups.) The trace, for instance, may not have the properties you are familiar with.

That being said, my only experience with octonionic matrices is the 27-dimensional exceptional Jordan algebra, described for example here. It consists of $3\times 3$ hermitian octonionic matrices and the Jordan product is given by the symmetrised product.

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