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 formula_1 and a subalgebra formula_2 reductive in formula_1. 
A reductive pair formula_4 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 formula_7 of formula_8, where
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 
formula_17
is the homotopy quotient, here homotopy equivalent to the regular quotient, and
Then the characteristic algebra is the image of formula_19, the transgression formula_20 from the primitive subspace "P" of formula_21 is that arising from the edge maps in the Serre spectral sequence of the universal bundle formula_22, and the subspace formula_9 of formula_24 is the kernel of formula_25.