The Stacks project

Sketch of proof. To prove that $\delta$ defines a group homomorphism, i.e., that $\delta (A \otimes _ K B) = \delta (A) + \delta (B)$, one computes directly with cocycles.

Injectivity of $\delta$. In the abelian case ($d = 1$), one has the identification

the latter of which is trivial by fpqc descent. If this were true in the non-abelian case, this would readily imply injectivity of $\delta$. (See [SGA4.5].) Rather, to prove this, one can reinterpret $\delta ([A])$ as the obstruction to the existence of a $K$-vector space $V$ with a left $A$-module structure and such that $\dim _ K V = \deg A$. In the case where $V$ exists, one has $A \cong \text{End}_ K(V)$.

For surjectivity, pick a cohomology class $\xi \in H^2(\text{Gal}(K^{sep}/K), (K^{sep})^*)$, then there exists a finite Galois extension $K^{sep}/K'/K$ such that $\xi$ is the image of some $\xi ' \in H^2(\text{Gal}(K'|K), (K')^*)$. Then write down an explicit central simple algebra over $K$ using the data $K', \xi '$. $\square$