The octonions play a significant role in the classification and construction of other mathematical entities. For example, the exceptional Lie group is the automorphism group of the octonions, and the other exceptional Lie groups , , and can be understood as the isometries of certain projective planes defined using the octonions. The set of self-adjoint 3 × 3 octonionic matrices, equipped with a symmetrized matrix product, defines the Albert algebra. In discrete mathematics, the octonions provide an elementary derivation of the Leech lattice, and thus they are closely related to the sporadic simple groups.