Can Principal Bundles Help with Lie Group Decomposition?

[Pond Dragon]

Long time reader, first time poster.

Originally, it was my contention that all Lie groups could be written as the semidirect product of a connected Lie group and a discrete Lie group. However, I no longer believe this is true.

The next best thing I could think of was to say that a Lie group is diffeomorphic to the (smooth) product manifold of a connected Lie group and a discrete one. This is proving to be difficult!

I'm a bit new to using general fiber bundles (I've only ever used vector bundles before!), but a friend of mine recommended me to use principal bundles. I've proven that $\pi:G\to G/G_\mathrm{e}$, where $G_\mathrm{e}$ is the identity component of a Lie group $G$, is a principal $G_\mathrm{e}$-bundle. However, I really don't know what use this is to me.

Could someone explain how this is a step in the right direction?

Edit: Is there a particular way to get the LaTeX to render? It isn't working...
Edit2: In another post, display math works. What about inline...?
Edit3: Crisis averted.

 

[Pond Dragon]

Clearly, I am an idiot.

To those who might come looking for this thread later: principal bundles are nice, but it is easier to simply note that the connected components of a Lie group are diffeomorphic.