Theorems about algebras

theorem that the finite-dimensional associative division algebras over the reals are either the reals, the complex numbers, or the quaternions

theorem stating that every finite-dimensional Lie algebra over a field is a Lie algebra of square matrices

explicitly describes the universal enveloping algebra of a Lie algebra

theorem on finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form

theorem representing a solvable Lie algebra as an algebra of upper triangular matrices; generalized by Lie–Kolchin theorem

theorem in the mathematics of Lie's theory