Posting this here and hoping that someone has come across it since I couldn't find an example anywhere online that looks similar.
I tried to find out how to convert this to strategic or normal form (which is the task I am supposed to do) but I am not completely sure on how I would do this considering that nature is present at a lower node.
My best guess so far is to include the part with nature and the probability of $1/2$ like this:
(a,a) (a,b) (b,a) (b,b)
A (1.5,3.5) (1.5,3.5) (1.5,3.5) (1.5,3.5)
before continuing with the usual. I tried to find a similar example online, but whenever nature is involved, it is introduced at the root not as a leaf node leading to immediate payoffs. Since I cannot find an example online of this case, I do not know what is convention in this case.
Any help would be greatly appreciated!
Your approach with Nature is correct: replace the terminal subgame where Nature plays with expected payoffs.
Be careful elsewhere, however: since Player 2 has an information set, she cannot tell apart $a$ from $b$ at different nodes, so her strategies are simply $a$ and $b$.