Cartan pair

In the mathematical fields of Lie theory and algebraic topology, the notion of Cartan pair is a technical condition on the relationship between a reductive Lie algebra and a subalgebra reductive in.
A reductive pair is said to be Cartan if the relative Lie algebra cohomology
is isomorphic to the tensor product of the characteristic subalgebra
and an exterior subalgebra of, where

, the Samelson subspace, are those primitive elements in the kernel of the composition,
is the primitive subspace of,
is the transgression,
and the map of symmetric algebras is induced by the restriction map of dual vector spaces.

On the level of Lie groups, if G is a compact, connected Lie group and K a closed connected subgroup, there are natural fiber bundles
where
is the homotopy quotient, here homotopy equivalent to the regular quotient, and
Then the characteristic algebra is the image of, the transgression from the primitive subspace P of is that arising from the edge maps in the Serre spectral sequence of the universal bundle, and the subspace of is the kernel of.