Rost Multipliers of Lifted Kronecker Tensor Products

We extend techniques employed by Garibaldi to construct various new injections involving the half-spin group, $\textbf{HSpin}$, induced by lifting the Kronecker tensor product to simply connected groups. We calculate the Rost multipliers of the maps we have constructed. Furthermore, we utilize our new map $\textbf{PSp}_{2n}\times \textbf{PSp}_{2m} \hookrightarrow \textbf{HSpin}_{4nm}$ to describe the structure of the normalized degree three cohomological invariants of $\textbf{HSpin}_{4n}$.