  Let $k$ be a field of characteristic not $2$. We give a positive answer to
Serre's injectivity question for any smooth connected reductive $k$-group whose
Dynkin diagram contains connected components only of type $A_n$, $B_n$ or
$C_n$. We do this by relating Serre's question to the norm principles proved by
Barquero and Merkurjev. We give a scalar obstruction defined up to spinor norms
whose vanishing will imply the norm principle for the non-trialitarian $D_{n}$
case and yield a positive answer to Serre's question for connected reductive
$k$-groups whose Dynkin diagrams contain components of non-trialitarian type
$D_n$ also. We also investigate Serre's question for reductive $k$-groups whose
derived subgroups admit quasi-split simply connected covers.
