Let $V$ be the set of prime together with the symbol $\infty$. For a prime $v=p$, denote the $p$-adic numbers by $\mathbb{Q}_p$ and the real numbers by $\mathbb{Q}_\infty$. For $v\in V$ the Hilbert symbol is defined for $a,b\in\mathbb{Q}^*_v$ as
\begin{align*} (a,b)_v=\begin{cases}+1,&\text{ if }ax^2+by^2=z^2\text{ has a non-zero solution }(x,y,z)\in \mathbb{Q}_v^3;\\-1,&\text{ else.}\end{cases} \end{align*}
Furthermore for $v\in V, a,b\in\mathbb{Q}^*$ we denote by $(a,b)_v$ the Hilbert symbol of $(\bar a,\bar b)_v$ where $\bar a,\bar b$ are the images of $a,b$ in $\mathbb{Q}_v$.
Now a theorem by Hilbert says that $(a,b)_v=1$ for almost all $v\in V$ (and that furthermore $\prod_{v\in V}(a,b)_v=1$, but I'm not interested in this at the moment). The theorem can be found in "A course in Arithmetic" by Jean-Pierre Serre for example.
It basically says that there is a finite set $E\subseteq V$ such that \begin{align*} (a,b)_v=\begin{cases}+1,&\text{ if }v\notin E\\-1,&\text{ if }v\in E\end{cases} \end{align*}
My question is if this set $E$ has a common name in the literature. Something like $E_{a,b}$ would make sense to me (since it depends on $a$ and $b$). If there is no widely used name, what are your suggestions?
I don't know of a common term for this set, but calling it a Hilbert set or something similar would be reasonable. Let me propose something only slightly fancier below.
Note that $(a,b)_v=-1$ if and only if the quaternion algebra $\langle a,b\rangle_\mathbb{Q}$ (among hundreds of other notations) is ramified at $v$, i.e., if $v$ divides its discriminant. So if you were in a setting where, say, algebraic geometry language was convenient, you might call this set the "Hilbert support of $\langle a,b\rangle$", or maybe reference the Hilbert radical if it were more convenient to refer to the product of such primes (which is occasionally useful).
Edit: I see that SAGE calls the Hilbert conductor what I call the Hilbert radical. That works too.