An example is the Siegel upper half plane, where V⊂Rk(k + 1)/2 is the cone of positive definite quadratic forms in Rk and m = k(k + 1)/2.
Élie Cartan classified the homogeneous bounded domains in dimension at most 3 (up to isomorphism), showing that they are all Hermitian symmetric spaces.
Suppose that G is the Lie algebra of a transitive connected group of analytic automorphisms of a bounded homogeneous domain X, and let K be the subalgebra fixing a point x.
Then the almost complex structure j on X induces a vector space endomorphism j of G such that A j-algebra is a Lie algebra G with a subalgebra K and a linear map j satisfying the properties above.
The converse is also true: any j-algebra is the Lie algebra of some transitive group of automorphisms of a homogeneous bounded domain.