One important property of the so called norm varieties defined by Rost, is that they split pure symbols of Galois cohomology groups $H^n(k,\mu_p)$:
For some $\alpha \in H^n(k,\mu_p)$ there is a smooth, projective $X_\alpha$ such that
$X_\alpha$ has a $L$-rational point iff $\alpha = 0$ in $H^n(L,\mu_p)$ for all extensions $L/k$.
We call such a variety a splitting variety for $\alpha$.
Lets assume we have non trivial, pure symbols $\alpha \in H^n(k,\mu_p)$ and $\beta \in H^m(k,\mu_l)$ for different primes $p,l$.
Is there an example of a variety $X$ which is a splitting variety for both symbols $\alpha, \beta$ i.e. is it possible to construct such a thing?
Is it possible when we restrict our definition to field extensions of dimension $p*l$ ?