If you are only concerned about commutative subalgebras of $M_n(\mathbb{C})$ then there is a fairly easy characterization. So any abelian algebra is generated by a single self adjoint element (spectral theorem). Call this element T. Then T is diagonalizable and so the algebra it form will be the algebra of polynomials over it. Since it is diagonalizable that is a unitarty $u$ with $uTu^*$ diagonal. And the algebra has dimension $k$ exactly when T has $k$ distinct non-zero eigenvalues.

Note: This is assuming that T is invertible. If T is not invertible then the polynomial algebra since it contains the constants will have dimension $k+1$.

So then we can view the algebra generated by T as an algebra of the form $u^*Au$ where A is an algebra of diagonal matrices.

If you are only concerned about commutative subalgebras of $M_n(\mathbb{C})$ then there is a fairly easy characterization. So any self-adjoint abelian algebra is generated by a single self adjoint element (spectral theorem). Call this element T. Then T is diagonalizable and so the algebra it form will be the algebra of polynomials over it. Since it is diagonalizable that is a unitarty $u$ with $uTu^*$ diagonal. And the algebra has dimension $k$ exactly when T has $k$ distinct non-zero eigenvalues.

Note: This is assuming that T is invertible. If T is not invertible then the polynomial algebra since it contains the constants will have dimension $k+1$.

So then we can view the algebra generated by T as an algebra of the form $u^*Au$ where A is an algebra of diagonal matrices.