Finiteness of Hermitian Levels of Some Algebras

We characterize the hermitian levels of quaternion and octonion algebrasand of an 8-dimensional algebra D over the ground field F, constructed using a weak version of the Cayley-Dickson double process.It is shown that all values of the hermitian levels of quaternion algebras with the hat-involution also occur as hermitian levels of D. We give some limits to the levels of the algebra D over some ground field.