Galois cohomology elements becoming trivial over $k(\mathrm{SB}(A))$ Assume we work over a field $k$ of characteristic zero. Let $A$ be a central simple algebra over $k$ of index $\mathrm{ind}(A) = 2$ and let $[A]$ denote its class in Br$(k)$. Let $\alpha \in$ $H^3(k,\mu_2)$. Let us consider the restriction map in Galois cohomology to the function field of the Severi-Brauer variety of $A$, which we denote by $\mathrm{res}_{k(\mathrm{SB}(A))/k}$.
Assume that $\alpha$ lies in the kernel of $\mathrm{res}_{k(\mathrm{SB}(A))/k}$.
Question: Does $[A]$ divide $\alpha$ ? (over $k$)
The degree of $H^3$ and the case $\mathrm{ind}(A) = 2$ shouldn't influence the outcome, but I am particularly interested in this situation. Also the opposite direction is trivial and I do not see why $[A]$ would not divide $\alpha$, since I think the kernel of the restriction is generated by $[A]$. But still I am not sure if this is all one needs.
 A: Yes, it is true.
One can apply the results of Peyre's paper ``Products of Severi-Brauer varieties and Galois cohomology".
To state the result to use: let $X_1,...,X_n$ be Severi-Brauer varieties over $k$. Let $H\subset \text{Br}(k)$ be the subgroup generated by $[X_1],...,[X_n]$. Then, the kernel of the restriction $$H^3(k,\mathbf{Q}/\mathbf{Z}(2))\rightarrow H^3(k(X_1\times\cdots\times X_n),\mathbf{Q}/\mathbf{Z}(2))$$ modulo the subgroup $H^1(k,\mu_2)\cup H$ is isomorphic with the torsion subgroup of codimension 2 cycles of the product of these varieties $\text{CH}^2(X_1\times \cdots \times X_n)_{tors}$.
Of course, in the situation you are interested, you are only considering $2$-torsion in the Chow ring of a conic. This is trivial because of the dimension. The whole proof for your claim is given its own section in the above paper.
In general though, contrary to your belief it shouldn't rely much on the index, there are central simple algebras $A$ of arbitrary degree which will have a large torsion subgroup in $\text{CH}^2(\text{SB}(A))$. For first results in this direction, one should consult the paper of Karpenko titled ``Codimension 2 cycles on Severi-Brauer varieties". Karpenko has a number of other papers computing torsion in related groups as well.
