Sophus Lie proved three important theorems about Lie groups and Lie algebras. The third theorem details the correspondence between the two and states that every finite-dimensional real Lie algebra is the Lie algebra of some simply connected Lie group. There is also a correspondence between homomorphisms of algebras and groups and between subalgebras and subgroups.

For example, let G denote the group of n×n matrices with positive determinant. This has Lie algebra M_n×n(ℝ), the space of n×n matrices with Lie bracket [A,B] = AB–BA. The exponential map exp:M_n×n(ℝ) → G, taking a matrix to its exponential, has an important role in Lie’s theory. Now the determinant map det: G → (0,∞) is a homomorphism of Lie groups, which must correspond to a homomorphism of the Lie algebras M_n×n(ℝ) → ℝ; in this case that homomorphism is trace. Further, as

is a commutative diagram, this means that