For a real Lie algebra, splittable is equivalent to either of these conditions:

There are 3 simple real Lie algebras associated with this root system:

Let be a real Lie algebra.

In the complex simple case, this is a classical result, whereas for real simple Lie algebras, this fact has been proven as recently as 2010.

This is simply a different, more convenient, representation of the same real Lie algebra.

Any Lie group G defines an associated real Lie algebra .

An important class of infinite-dimensional real Lie algebras arises in differential topology.

This is a number between 0 and the dimension of g which is an important invariant of the real Lie algebra, called its index.

The real exceptional Lie algebras appearing here can again be described by their maximal compact subalgebras.

However, there are additional "reality" structures that are invariant under the action of the real Lie algebras.