When is an algebra of commuting matrices (contained in one) generated by a single matrix?

Thanks for the answers. Just to wrap up a bit, here are a few examples.

1) Sometimes an ACM (algebra of commuting matrices) is sure to be generated by one of its members

2) Other times it has dimension too large to possibly be (embedded in) an ACM with a single generator.

3) An ACM might be generated by 2 matrices, not generated by any of its members, but embed in a larger ACM which does have a single generator.

4) An ACM might be generated by 2 matrices, not generated by any of its members, but not embed in a larger ACM which does have a single generator (even in the 3x3 case).

1) If the matrices are all normal then they can be simultaneously diagonalized. This reduces the problem to an algebra of diagonal matrices, which is easy to understand. Such an algebra is actually generated by one of its members.

2) The 5 dimensional algebra ${cal{A}}_5$ mentioned by Mariano (4x4 matrices with 2x2 blocks $left(begin{smallmatrix}0&A\0&0end{smallmatrix}right)$ has dimension too large to be generated by a single matrix. Furthermore, each member M generates only the 2 dimensional algebra of matrices $jI+kM$ so no subalgebra of dimension 3 or 4 has a single generator.

3) Consider the subalgebra ${cal{A}}_3$ of ${cal{A}}_5$ generated by $left(begin{smallmatrix}0&A\0&0end{smallmatrix}right)$ and $left(begin{smallmatrix}0&B\0&0end{smallmatrix}right)$ with A and B 2x2 invertible matrices (neither a scalar multiple of the other). As mentioned, we can't embed ${cal{A}}_3$ in a singly generated 4 dimensional subalgebra of ${cal{A}}_5$
However it also embeds in other 4 dimensional algebras.

For example a 4x4 matrix $left(begin{smallmatrix}BA^{-1}&C\0&A^{-1} Bend{smallmatrix}right)$ will generate an ACM which commutes with everything in ${cal{A}}_3$. I guess in this case it would automatically contain ${cal{A}}_3$. I certainly verified that randomly filling in the C does this in several cases. In many cases I tested one can get away with one or both of A and B having rank 1... but not always. The two 4x4 matrices made from matrices with $A=left(begin{smallmatrix}1&0\0&0end{smallmatrix}right)$ and $B=left(begin{smallmatrix}0&1\0&0end{smallmatrix}right)$ give an example of that. One can shrink this to 3x3 (in the case that the underlying ring is $mathbb{Z}_2$ as noted by Martin and missed by me), so I will:

4) The two $3 times 3$ matrices $left(begin{smallmatrix}0&1&0\0&0&0\0&0&0end{smallmatrix}right)$ and $left(begin{smallmatrix}0&0&1\0&0&0\0&0&0end{smallmatrix}right)$ generate an algebra A with eight members $left(begin{smallmatrix}a&b&c\0&a&0\0&0&aend{smallmatrix}right)$.
A is maximal but not generated by any of its members as each member generates a 2 dimensional (or smaller) subalgebra.