The equation shown is a type of decomposition of a Lie algebra g that was introduced by Élie Cartan in his doctoral thesis. h is called a Cartan subalgebra of g.

To answer what a Lie algebra is, an extremely hand-wavy description might be: a Lie group is a continuous group of symmetries of some geometric object (for example, the group SO(3) of rotations of three-dimensional space), and the corresponding Lie algebra is the "tangent space" to the group, that is, the space of tiny changes you can make that lie "along" the group.

A lot of things in Lie theory and differential geometry are named after Élie Cartan, or sometimes after his son Henri.