For a commutative ring $R$ satisfying the condition $\mathrm{sr}(R)\leq n$ and a root system $\Phi_n$ of type $B_n$ or $C_n$ we compute the kernel of the stabilization map $\mathrm{K}_1(\Phi_n, R)\righarrow\mathrm{K}_1(\Phi_{n+1}, R)$.