Let $R$ be a commutative unital ring. For each $n \ge 1$, let $GL_n(R)$ be the group of invertible $n \times n$ matrices over $R$, let $GL(R)=\varinjlim_n GL_n(R)$ under the stabilization maps $A\mapsto \operatorname{diag}(A,1_R)$, and let $K_1(R)$ be the stable Whitehead group of $R$. Let