# Albert algebra

In mathematics, an **Albert algebra** is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there are three such Jordan algebras up to isomorphism.^{[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

Over any algebraically closed field, there is just one Albert algebra, and its automorphism group *G* is the simple split group of type F_{4}.^{[2]}^{[3]} (For example, the complexifications of the three Albert algebras over the real numbers are isomorphic Albert algebras over the complex numbers.) Because of this, for a general field *F*, the Albert algebras are classified by the Galois cohomology group H^{1}(*F*,*G*).^{[4]}

The Kantor–Koecher–Tits construction applied to an Albert algebra gives a form of the E7 Lie algebra. The split Albert algebra is used in a construction of a 56-dimensional structurable algebra whose automorphism group has identity component the simply-connected algebraic group of type E_{6}.^{[5]}

The space of cohomological invariants of Albert algebras a field *F* (of characteristic not 2) with coefficients in **Z**/2**Z** is a free module over the cohomology ring of *F* with a basis 1, *f*_{3}, *f*_{5}, of degrees 0, 3, 5.^{[6]} The cohomological invariants with 3-torsion coefficients have a basis 1, *g*_{3} of degrees 0, 3.^{[7]} The invariants *f*_{3} and *g*_{3} are the primary components of the Rost invariant.