Actually I'm not sure about the term "commuting algebra". I might have used it wrongly.
Let $A$ be a non-scalar $2 \times 2$ matrix. Consider the ring of all $2 \times 2$ matrices which commute with $A$. We can show this ring is commutative!
This result is related to this question: http://mathoverflow.net/questions/13349/tate-module-of-cm-elliptic-curves
I just did this exercise by down-to-earth linear algebra computation. I'm wondering if there is a more conceptual proof, and is it still true for $n>2$?