  Under the action of the Geroch group, the Minkowski metric can be transformed
into any vacuum metric with two commuting Killing vectors. In principle, this
reduces the problem of deriving vacuum metrics with two commuting Killing
vectors to pure algebra. In this short note, we use these facts to give a
purely algebraic derivation of the Einstein-Rosen metric, which describes a
cylindrical gravitational wave. Our derivation has a straightforward extension
to gravitational pulse waves.
