  We show that the quantum Frobenius morphism constructed by Lusztig in the
setting of the quantum enveloping algebra specialized at a root of unity admits
a multiplicative splitting (non unital). We also find a basis of the toral part
of the small quantum algebra consisting of pairwise orthogonal idempotents
summing up to 1, and likewise in the modular case of the algebra of
distributions for a semisimple algebraic group.
