They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers.
[1] One of them, which was first mentioned by Pascual Jordan, John von Neumann, and Eugene Wigner (1934) and studied by Albert (1934), is the set of 3×3 self-adjoint matrices over the octonions, equipped with the binary operation where
The final is constructed from the non-split octonions using a different standard involution.
Because of this, for a general field F, the Albert algebras are classified by the Galois cohomology group H1(F,G).
[7] The space of cohomological invariants of Albert algebras a field F (of characteristic not 2) with coefficients in Z/2Z is a free module over the cohomology ring of F with a basis 1, f3, f5, of degrees 0, 3, 5.