The set
$$S= \left\{ \left(\begin{array}{cc}a&b \\ -b&a \end{array}\right):a, b\in\mathbb{R} \right\}$$
is a subring of the matrix ring $M_2(\mathbb R)$ isomorphic to $\mathbb C$. Can we find other subring $L$ of $M_2(\mathbb R)$ which is also isomorphic to $\mathbb C$ and such that $S\cap L= \left\{ \left(\begin{array}{cc}a&0 \\ 0&a \end{array}\right):a \in\mathbb R\right\}$?