  We give a short and purely bilinear proof of the fact that two chains of
$p$-elementary lattices with quadratic form or alternating bilinear form over
the $p$-adic integers ore more generally over a complete discrete valuation
ring have common hyperbolic bases. This fact, which is useful for the study of
Bruhat-Tits buildings, has been proven before with different methods by
Abramenko and Nebe and by Frisch.
