Steinberg modules and Donkin pairs

First we prove that in characteristicp>0 a module with good filtration for a group of type E₆ restricts to a module with good filtration for a group of type F₄. Thus we confirm a conjecture of Brundan for one more case. Our method relies on the canonical Frobenius splittings of Mathieu. Next we settle the remaining cases, with a computer-aided variation on the old method of Donkin.