Dimension of extension group is sum of dimensions of normal subgroup and quotient group

Statement for algebraic groups
Suppose $$G$$ is an algebraic group, $$N$$ is a closed normal subgroup and $$G/N$$ is the quotient group. Both $$N$$ and $$G/N$$ acquire algebraic group structures from the algebraic group structure of $$G$$. We then have the following relationship between the dimensions of $$G,N,G/N$$:

$$\dim G = \dim N + \dim (G/N)$$

In particular, $$G$$ is a finite-dimensional algebraic group if and only if both $$N$$ and $$G/N$$ are.

Statement for Lie groups
Suppose $$G$$ is a Lie group, $$N$$ is a closed normal subgroup and $$G/N$$ is the quotient group. Both $$N$$ and $$G/N$$ acquire Lie group structures from the Lie group structure of $$G$$. We then have the following relationship between the dimensions of $$G,N,G/N$$:

$$\dim G = \dim N + \dim (G/N)$$

Related facts

 * Dimension of direct product is sum of dimensions