Families of lattices with an unbounded number of unit vectors

3 families of 4-dimensional lattices $L_k,M_k,M_k\mathrm{/}2\subset {\mathbb{R}}^2$ are defined. Each lattice is defined by 2 quadratic extensions and has a \emph{finite} number of unit vectors, but the number of unit vectors in each of the 3 familes is \emph{unbounded}. $L_3$ is the Moser lattice.