  For central simple algebras of exponent $2$ over fields of characteristic $2$
and $2$-cohomological dimension equal to $2$, we study the adapted
decomposition to some multiquadratic extensions of the base field. Several
remarkable properties are extended to multiquadratic extensions of separability
degree at most $4$. We also extend to the characteristic $2$ a result of
Elman-Lam-Tignol-Wadsworth by constructing an algebra of exponent $2$ and
degree $8$ containing a separable triquadratic extension but which admits no
adapted decomposition to this extension. As an application we give an
elementary proof of the non-excellence of separable biquadratic extensions.
