Lie algebras of the unit group of a finite dimensional algebra

Question

Given a finite dimmensional associative algebra $A$, it can be proven that the unit group $$A^{\ast}:=\{a\in A \mid \exists\, b \in A \mbox{ such that } ab=ba=1_A\} $$ is always a Lie group (The unit group of a finite dimensional associative algebra is a Lie group?), but I have problems trying to identify its Lie algebra. I think that its Lie algebra $(\mathfrak{a},[.,.])$ should be isomorphic (as Lie algebras) to $(A,[.,.]_c)$, where $[a,b]_c=ab-ba$ (as the case $A=M_n(\mathbb{K})$). It is true?? I hope somebody can help me