This is a continuation of a previous question: Connectedness of groups of units in finite-dimensional commutative algebras.

Let $k$ be an algebraically closed field of characteristic $0$. Which algebraic groups / Lie groups (for $k=\mathbb{C}$) can occur as the groups of units of finite-dimensional noncommutative associative $k$-algebras $A$?

Even though a classification of finite-dimensional *commutative* $k$-algebras seems (at the moment) out of reach, it turned out that their groups of units are easy to understand (see the linked question). So I am hoping the answer to this new question might also be feasible despite the fact that a classification of finite-dimensional *noncommutative* $k$-algebras is hopeless.

Obviously, $U(A)$ has to contain a copy of $\mathbb{G}_m$, which automatically excludes all compact Lie groups. Moreover, the following remark of Tom Goodwillie applies:

"Even without assuming commutativity, the set of non-units is the vanishing set of a not-identically-zero complex polynomial function, the function sending $x$ to the determinant of $y\mapsto xy$. This has complex codimension $\ge 1$, so its complement is connected." (answer at Oct 28 '15 at 18:41 in op. cit.)