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.



Lorentzian characterization of genus
A good reference for this assertion is Cassels's "Rational Quadratic Forms", though you have to dig a bit. Let me see if I can outline the proof. First, I think Conway and Sloane assume $f$ and $g$ are classical integral (i.e. correspond to even lattices). In my copy of SPLAG, at the end of subsection 2.1 of that chapter, they say "so in this book we call $f$ an integral form if and only if its matrix coefficients are integers (i.e. if and only if it is classically integral ...)".


Now suppose $f$ and $g$ are in the same genus. Then so are $f\oplus H$ and $g \oplus H$. Next, we want to show they're in the same spinor genus. This follows from the Corollary of Lemma 3.6 of Chapter 11 of Cassels: "If we show $U\_p \subset \theta(\Lambda\_p)$ for all $p$, then the genus of $\Lambda$ consists of a single spinor genus". Here $\Lambda = f \oplus U$, where I'm identifying the form and the lattice by a bit of abuse of notation. Since $\theta(\Lambda\_p) \supset \theta(H\_p)$ (see a few sentences below the corollary), and $\theta(H\_p) \supset U\_p$ by Lemmas 3.7 and 3.8, we've proved that the genus consists of a single spinor genus.


Finally, since the forms are indefinite of dimension at least $3$, the spinor genus consists of a single class.


To go back is the easier direction (I think): if $f \oplus U$ is equivalent to $g \oplus U$, then they are equivalent over $\mathbb{Z}\_p$ for every $p$. Then an analogue of Witt cancellation will do the job (see Chapter 8 of Cassels).


