The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple).
Column legend with Lie bracket the cross product; also isomorphic to su(2) and to so(3,R) but not simple.
Note that a "complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given by a holomorphic map.
where J is the standard skew-symmetric matrix The Lie algebra of affine transformations of dimension two, in fact, exist for any field.
An instance has already been listed in the first table for real Lie algebras.