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 $\mathrm{GL}(n,R) \to \mathrm{GL}(n+1,R)$ (which add a unit vector as new last row and column) yields, by definition, the stable linear group $\mathrm{GL}(R)$; 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 $K_1(R) = \mathrm{GL}(R)/E(R)$ of $R$, giving an elementary description that does not involve elementary matrices explicitly.