[Niveau hermitien de Certaines algèbres de quaternions]

Let D = (a,b/F) a quaternion divisor algebra over a Field F of characteristic not equal 2. Denote l, i, j, k the basis of D, such that i(2) = a, j(2) = b, ij = -ji = k and boolean AND : D --> D the involution given by (i)over cap = -i, (j)over cap = j (and (k)over cap = k). In [LE] D. LEWIS asks the following question : Does there exist a quadratic Pfister form [S p. 72] phi such that the hermitian form phi x D is isotropic over (D, boolean AND) but not hyperbolic? In this note, we show that the answer of this question is negative, so that the hermitian level [ I], when it is finite, of (D, boolean AND) is a power of two. This result holds for quaternion algebras with standard involution [LE].