  We give a unified description of tetrahedra with lightlike faces in 3d
anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de
Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra
are determined by a generalized cross-ratio with values in a commutative 2d
real algebra that generalizes the complex numbers. Equivalently, tetrahedra
with lightlike faces are determined by a pair of edge lengths and their duals
by a pair of dihedral angles. We prove that the dual tetrahedra are precisely
the generalized ideal tetrahedra introduced by Danciger. Finally, we compute
the volumes of both types of tetrahedra as functions of their edge lengths or
dihedral angles, obtaining generalizations of the Milnor-Lobachevsky volume
formula of ideal hyperbolic tetrahedra.
