On the decomposability of bilinear spaces of dimension four

In this paper, we study the problem of decomposability of bilinear spaces of dimension four without symmetry, as well as the problem of decomposability of split central simple algebras of degree four with an anti-automorphism. In particular, we show that, contrary to the case of symmetric or skew-symmetric bilinear spaces, these two problems are not equivalent. We will also prove that cohomological invariants do not detect decomposability of bilinear spaces of dimension four in general, whereas the determinant does for split central simple algebras of degree four with an anti-automorphism.