A structure theorem for euclidean buildings

We prove an affine analog of Scharlau’s reduction theorem for spherical buildings. To be a bit more precise let $X$ be a euclidean building with spherical building $\mathrm{\partial }X$ at infinity. Then there exists a euclidean building $\bar{X}$ such that $X$ splits as a product of $\bar{X}$ with some euclidean $k$-space such that $\mathrm{\partial }\bar{X}$ is the thick reduction of $\mathrm{\partial }X$ in the sense of Scharlau. \newline In addition we prove a converse statement saying that an embedding of a thick spherical building at infinity extends to an embedding of the euclidean building having the extended spherical building as its boundary.