Does there exist a partition of an L to create a square?

If $a$ is small enough, a well-known dissection used to prove Pythagoras' theorem can serve as a basis to create a partition of the L (see diagram below). Of course things become more complicated if $a$ gets bigger, I haven't tried that case to see if one can obtain a simple partition.

EDIT.

Here is another partition inspired by a Pythagorean proof (# 26 at https://www.cut-the-knot.org/pythagoras/). This one allows larger values of $a$.

It can be done. Whether or not it can be done nicely depends on your definition of "nicely".

In geometry, the Wallace–Bolyai–Gerwien theorem, named after William Wallace, Farkas Bolyai and Paul Gerwien, is a theorem related to dissections of polygons. It answers the question when one polygon can be formed from another by cutting it into a finite number of pieces and recomposing these by translations and rotations. The Wallace-Bolyai-Gerwien theorem states that this can be done if and only if two polygons have the same area.

https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem