  This paper is dedicated to the surjective stability of the $K_1$-functor for
Chevalley groups for the embedding $D_5$ into $E_6$. This case was already
studed by Plotkin. In the present paper, we improve his result by showing that
surjective stability holds under a weaker assumption on a ring. Another result
of the present paper shows how the $K_1$-stability can help to study overgroups
of subsystem subgroups.
