Let \(F\) be a field of characteristic different from \(2\), \(\mu _ 2\) the subgroup \(\langle -1\rangle \) of the multiplicative group \(F ^ {\ast }\), and Sets, Rings, Alg\(_ F\) and Fields\( _ F\) the categories of sets, rings, commutative \(F\)-algebras, and field extensions of \(F\), respectively. For each integer \(n \geq 1\), denote by \({\mathbf{CSA}}_ n\colon{\mathbf{Alg}}_ F \to {\mathbf{Sets}}\) the functor of isomorphism classes of Azumaya algebras of rank \(n\), and by { MS}\(_ {n,r}\) the subfunctor of CSA\(_ {n ^ r}\) of isomorphism classes of tensor products of \(r\) symbol algebras of degree \(n\), for any integer \(r \geq 1\). Let \(H ^ {\ast }(K)\) be the cohomology ring of \(K\) with coefficients in \(\mu _ 2\), for any field extension \(K/F\). The correspondence \(K \to H ^ {\ast }(K)\) gives rise to a functor \(H ^ {\ast }\colon\) Fields\(_ F \to\) Rings. Given a covariant functor \({\mathbf F}\colon\) Fields\(_ F \to\) Sets, by a cohomological invariant of F over \(F\), we mean a natural transformation F\( \to H ^ {\ast }\) of functors Fields\(_ F\) from Sets. The set of these invariants is denoted by Inv(F,\(H ^ {\ast })\) and can be canonically viewed as an \(H ^ {\ast }(F)\)-module. The paper under review proves an unpublished theorem due to Rost, which describes the cohomological invariants of central simple algebras of degree \(4\) with values in \(\mu _ 2\), under the hypothesis that the base field \(F\) contains a square root of \(-1\). This theorem proves the injectivity of a certain mapping of Inv\(({\mathbf{CSA}}_ 4, H ^ {\ast })\) into Inv\(({\mathbf{MS}}_ {2,2}, H ^ {\ast }) \times {\text{ Inv}}({\mathbf{MS}}_ {4,1}, H ^ {\ast })\). It implies that Inv\(({\mathbf{CSA}}_ 4, H ^ {\ast })\) is a free \(H ^ {\ast }(F)\)-module with a certain explicitly defined basis. Also, it shows that a cohomological invariant is identically zero, provided that it is zero on biquaternion algebras and cyclic algebras. The proof relies on the Rost-Serre-Tignol theorem on the decomposition of the trace quadratic form of any central simple \(F\)-algebra in the Witt group of \(F\) see [M. Rost, J.-P. Serre and J.-P. Tignol, C. R., Math., Acad. Sci. Paris 342, No. 2, 83–87 (2006; Zbl 1110.16014)].