A seminorm with square property on a Banach algebra is submultiplicative

The result stated in the title is proved in a Banach algebra and is used to discuss (i) commutativity criteria in normed algebras, (ii) uniqueness of the uniform norm in uniform Banach algebras, and (iii) existence of continuous multiplicative linear functionals on topological algebras together with a simple reduction of the Michael problem in Frechet algebras. Submultiplicativity does not imply subadditivity in the presence of the square property