Electrons make transition between different energy levels - say, $n =3$ to $n=2$ or $n =1$, as per the applicable selection rules.
My question is: can an electron make transitions between sub energy states of the same energy level?
For example: if an electron is in the $n=3$ energy level, can it make a transition between different sub energy states of $n=3$ following the selection rules?
It depends on whether you're considering only hydrogen, or whether you're considering multi-electron atoms.
In hydrogen the states with different $\ell$ are (very nearly) degenerate in energy, so a transition like $2s^1 \rightarrow 2p^1$ is in principle allowed but in practice unobservable. However in Helium the $2s$ and $2p$ levels are not degenerate and indeed the transition line for $1s^12s^1 \rightarrow 1s^12p^1$ can be observed and has a wavelength of about a micron.
