Noncommutative rings with abelian group of units

Let $k$ be a field and $k[X,Y]$ be the ring of non-commutative polynomials over $k$. The invertible elements are exactly the elements of $k$.

My thinking about this question went as as follows:

1) When I think of non-commutativity, one of the first things that comes to my mind are matrices.

2) I want a ring of matrices whose unit group is nice (thinking: finite of small order).

3) What about the non-commutative ring $R\,$ of $2\times 2$ matrices with entries in $\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z}.$

4) By definition $R^\times = GL_2(\mathbb{F}_2)$ and a simple counting argument shows that $\#GL_2(\mathbb{F}_2) = 6.$

5) There are 2 groups of order 6 up to isomorphism: $C_6$ (abelian) and $S_3$ (non-abelian).

6) A little messing around showed me that $GL_2(\mathbb{F}_2)$ is not abelian :(

7) If $R^\times$ were smaller, then it would have to be abelian because all groups of order $<6$ are abelian.

8) Let's try to find a non-commutative subring of $R.$

9) What about the subring $S$ of all upper triangular matrices.

10) The unit group consists of all upper triangular matrices which have non-zero diagonal entries.

11) There are exactly two such matrices and so $\#S^\times = 2$ and we're done! :)

I hope this helps :)