An elementary description of $K_1(R)$ without elementary matrices

Let $R$ be a ring with unit. Passing to the colimit with respect to the standard inclusions $GL(n,R)\to GL(n+1,R)$ (which add a unit vector as new last row and column) yields, by definition, the stable linear group $GL(R)$; the same result is obtained, up to isomorphism, when using the “opposite” inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic $K$-group $K_1(R)=GL(R)\mathrm{/}E(R)$ of~$R$, giving an elementary description that does not involve elementary matrices explicitly.