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

All exact Borel subalgebras and all directed bocses are normal