Volumes of $L$-simplexes of five-dimensional lattices

We investigate the question of nonbasic Simplexes of an $L$-subdivision of five-dimensional lattices. It is shown that, apart from the Simplexes of volume $2V$ ($V$ is the volume of the basic simplex), no other nonbasic $L$-simplexes exist in these lattices. In primitive lattices the $L$-simplexes of doubled volume abut the basic $L$-simplexes by 4-dimensional faces.