Centralizer of a nonscalar two by two matrix ($AB=BA⟺B=\alpha I+\beta A$)

For any real nonscalar $2\times 2$ matrix $A$, its commuting matrices are exactly $B=\alpha I+\beta A$ with $\alpha ,\beta \in \mathbb{R}$. There is a vector $v$ for which $v,Av$ form a basis: otherwise every nonzero vector would be an eigenvector, and applying this to two independent vectors and their sum would force $A$ to be scalar. Given a commuting $B$, uniquely write $Bv=\alpha v+\beta Av$. Then $BAv=ABv=(\alpha I+\beta A)Av$, so $B$ and $\alpha I+\beta A$ agree on a basis and are equal. Conversely every polynomial in $A$ commutes with it. This generalizes the centralizer of a planar quarter-turn; the scalar-matrix exception is essential, because every matrix commutes with a scalar matrix.