Here, \vec{\xi}_2, \, \vec{\xi}_3, \, \vec{\xi}_4 generate the Euclidean group, acting within each planar wavefront, which justifies the name plane wave for this solution. Also \vec{\xi}_5, \, \vec{\xi}_6 show that all non transverse directions are equivalent. This corresponds to the fact that in flat spacetime, two colliding plane waves always collide head-on when represented in the appropriate Lorentz framework.
