In a two-factor design, a design which is optimal for one factor is shown to be optimal jointly for both the factors with respect to each of $A$-, $D$-, and $E$-optimalities. As an interesting consequence we have that a linked block design is optimal for the estimation of treatment differences. Similar results are also obtained for a class of three factor designs.
"Optimality of Two and Three Factor Designs." Ann. Statist. 4 (2) 419 - 422, March, 1976. https://doi.org/10.1214/aos/1176343420