The identity component of a subgroup has the same Lie algebra.

The identity component of this group is quadrant I.

The identity component of a topological group is always a characteristic subgroup.

Its identity component has the structure .

Each such subgroup is the identity component of the centralizer of its center.

Let K be a closed subgroup of G lying between G and its identity component.

The image of the exponential map always lies in the identity component of .

When is compact, the exponential map is surjective onto the identity component.

The radical of an algebraic group is the identity component of its maximal normal solvable subgroup.

The result is that the four quadrants are mapped into one, the identity component: