Let $SL_2(K)$ be the special linear group of rank 2 over a field $K$, as an affine group scheme cut out by the equation $ad - bc = 1$ in $\mathbb{A}^4_K$.
Let $H$ be an arbitrary, homogeneous, hyperplane, which contains the identity $I$ of $SL_2(K)$ as a point. For example, take $H: 2c - d + a = 0$. (I am particularly interested in this hyperplane).
Is it possible for $H$ to contain an entire conjugacy class besides that of the identity?