  As a step to establish the blockwise Alperin weight conjecture for all finite
groups, we verify the inductive blockwise Alperin weight condition introduced
by Navarro--Tiep and Sp\"ath for simple groups of Lie type $\mathsf A$, split
or twisted. Key to the proofs is to reduce the verification of the inductive
condition to the isolated (that means unipotent) blocks, using the Jordan
decomposition for blocks of finite reductive groups given by Bonnaf\'e, Dat and
Rouquier.
