The spectrum of units of algebraic $K$-theory

It is well known that the $[0,1]$ and $[0,2]$ Postnikov truncations of the units of the topological $K$-theories $\text{glone}\text{KO}$ and $\text{glone}\text{KU}$, respectively, are split, and that the splitting is provided by the ($\mathbb{Z}\mathrm{/}2$-graded) line bundles. In this paper we give a similar splitting for the $[0,1]$-truncation of the units of algebraic $K$-theory, considered as a sheaf on affine schemes. A crucial step is to produce the splitting for $\text{glone}K(\mathbb{Z})$. Along the way we also give a complete calculation of the connective spectrum of strict units of $K(\mathbb{Z})$ and $K({\text{F}}_{\mathrm{\ell }})$ for a prime $\mathrm{\ell }$. Finally, we show that the units of algebraic $K$-theory do not split as a presheaf. In fact we show they do not even split pointwise.