Suppose we take the "even" indefinite lattice from page 50 in Serre *A Course in Arithmetic* (1973)
$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right),$$
called $H$ in pages 189-191 of Larry J. Gerstein *Basic Quadratic Forms*.

What I cannot find in any detail is a proof of this arithmetic statement in
*SPLAG* by Conway and Sloane, page 378 in the first edition(1988), anyway
chapter 15 section 7, that quadratic forms $f,g$ are in the same genus
if and only if $f \oplus H$ and $g \oplus H$ are integrally equivalent. Then
they say this follows from properties of the spinor genus, presumably
including Eichler's theorem that indefinite rank at least 3 means
spinor genus and class coincide.
Also, if f and g do not correspond to "even lattices," I'm not
entirely sure what is being claimed. Oh, I absolutely cannot assume $f,g$ are in any way "unimodular." Very popular, that unimodular. Matter of taste, though. I'm not sure it matters, but my $f,g$ are going to be positive, which is surely the difficult case here.

Everybody with whom I have discussed this regards this as either
obvious or, essentially, an axiom. I would very much like a reference
for this, plus an explanation of what is meant if $f,g$ correspond to
"odd" lattices. For example, it would be wonderful if somewhere this claim and the words Theorem or Proposition or Lemma happened in the same sentence. I think I am making progress on the other bits I
need, essentially ch. 26,27 in *SPLAG*, but this claim has me snowed,
or perhaps buffaloed, thrown, stumped. As far as books that I own, I do not see the claim being discussed in Jones, Watson, O'Meara, Serre, Cassels, Kitaoka, Ebeling, Gerstein. I stopped by the office of R. Borcherds and discussed related matters for a while, the relevant articles are 1985 *The Leech Lattice* and 1990 *Lattices Like the Leech Lattice*, but I don't see the SPLAG claim in an explicit manner.

EDIT... Sexy application: the Leech lattice and all the Niemeier lattices are in the same genus. Pointed out in an MO comment by Noam Elkies, who knows things.