La forme trace d'une algèbre simple centrale de degré 4

Let k be a field of characteristic different from 2 containing a primitive 4-th root of unity. We show that the trace quadratic form of any central simple k-algebra A of degree 4 decomposes in the Witt group of k as the sum of a 2-fold Pfister form q_2 and a 4-fold Pfister form q_4 which are uniquely determined by A. The form q_2 is the norm form of the quaternion algebra Brauer-equivalent to A^2, and q_4 is hyperbolic if and only if A is a symbol algebra