  Recently, Brzezi\'nski, Koenig and K\"ulshammer have introduced the notion of
normal exact Borel subalgebra of a quasihereditary algebra. They have shown
that there exists a one-to-one correspondence between normal directed bocses
and quasihereditary algebras with a normal and homological exact Borel
subalgebra. In this short note, we prove that every exact Borel subalgebra is
automatically normal. As a corollary, we conclude that every directed bocs has
a group-like element. These results simplify Brzezi\'nski, Koenig and
K\"ulshammer's bijection.
