Obstruction of existing a 2-cocycle in Galois cohomology with given local components

Question

Let $K$ be a global field and $T$ be a torus which satisfy Hesse principle defined over $K$ and spilt over $L$. By tate-nakayama theorem we now that at every palace $l$,a cocharcter $\mu_l$ defined over $K$ defines a 2-cocycle $\xi_{\mu_l}$ in $H^2(K_l,T)=H^2(Gal(K_l^{al}/K_l),T(K_l^{al}))$. I want to know what is the obstruction for existing a 2-cocycle $\xi\in H^2(K,T)$ that goes to $\xi_{\mu_l}$ at each palace $l$. I think the obstruction is that the sum $$\sum_\ell \frac{1}{[L_\ell:K_\ell]}\sum_{\sigma\in Gal(L/K)}\sigma\mu_\ell=0$$ should be zero but I can't find any reference in the litreature. does someone know a reference for this?