Let $R$ be an associative unital ring. For each $n \ge 1$, let $GL_n(R)$ be embedded in $GL_{n+1}(R)$ by $A \mapsto \operatorname{diag}(A,1_R)$, and let $GL(R)=\varinjlim_n GL_n(R)$ be the stable general linear group with respect to these stabilization maps. For $n \ge 2$, $1 \le i \ne j \le n$, and $r \in R$, let $E_{ij}\in M_n(R)$ denote the matrix unit and let $e_{ij}(r)=I_n+rE_{ij}\in GL_n(R)$ be the elementary transvection, viewed as an element of $GL(R)$ under the stabilization maps. Let $E(R)\le GL(R)$ be the subgroup generated by all such elementary transvections. Then $E(R)\trianglelefteq GL(R)$.