Cassels proved that projectively equivalent integral quadratic forms are commensurable. In this note, an elementary proof of the converse of this theorem, for indefinite forms, is given. This was proved in "On integral quadratic forms having commensurable groups of automorphisms," Hiroshima Math. J. 43, 371–411 (2013) for forms of Sylvester signature +++. . .+- or ---. . .-+ (hyperbolic forms) and it was left there, as an open problem, for non-hyperbolic indefinite forms of any Sylvester signature.
"Addendum to ‘‘On integral quadratic forms having commensurable groups of automorphisms’’." Hiroshima Math. J. 44 (3) 341 - 350, November 2014. https://doi.org/10.32917/hmj/1419619751