Let $G$ be a topological group. One general approach to show that $G$ is connected is the following:

For every subgroup $H\leq G$ (not necessarily closed) we have a projection map: $$ \pi: G\rightarrow G/H $$ We may give to the set of left cosets of $H$ in $G$ the quotient topology. In that case $\pi$ is an open map. Now it is easy to see that if $H$ and $G/H$ are connected then $G$ is connected. Using this idea one may show that many of the classical groups are connected. For example, one may show that $U(n)$ is connected using induction and the usual fibration: $$ U(n-1)\rightarrow U(n)\rightarrow S^{2n-1} $$

Here is my question:

Q: Is there another general approach to show that a topological group $G$ is connected which avoids this kind of "devissage"?

(P.S. Of course one may try to prove directly that the space is path connected but in general this is too difficult)

(P.S.S. In the case of a real Lie group, using Cartan's theorem, one sees that the connectedness of $G$ is equivalent to the connectedness of a maximal compact subgroup of $G$, but I guess that this is a very special situation which does not generalize well to topological groups.)