All exact Borel subalgebras and all directed bocses are normal

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