Independent quaternion classes over rational numbers (${⨁}_{p\equiv 3(4)}\mathbb{Z}/2\hookrightarrow \mathrm{Br}(\mathbb{Q})$)

Associate to each prime $p\equiv 3(\mathrm{mod}4)$ the class of $(-1,p{)}_{\mathbb{Q}}$. These classes have order two, since quaternion conjugation identifies the algebra with its opposite. The product rule $[(-1,a)][(-1,b)]=[(-1,ab)]$ follows by factoring their tensor product into the commuting subalgebras generated by $i_1,j_1{j}_2$ and by $i_1{i}_2,j_2$; the second is the split algebra $(1,b)$. Every nonempty finite product of distinct such primes has an odd valuation at one of them. The quaternion norm obstruction at primes congruent to three modulo four therefore makes its class nonzero, proving independence. Euclid's construction $4p_1\cdots p_m-1$ proves there are infinitely many such primes.