In Ehrhart theory, the second coefficient of an Ehrhart polynomial for a polytope P in a lattice L can be computed by considering the following: "The lattice L induces a lattice $L_F$ on any face F of P; take the (d-1)-dimensional volume of F, divide by $2d(L_F)$ and add those numbers for all faces of P."
My question is what does it mean that the lattice L "induces" a lattice $L_F$? What would be the interpretation of this in the three-dimensional case if the lattice is $\mathbb{Z}^3$? I can't quite figure out how to actually compute this in a practical example. Any help is highly appreciated! Thank you.