Absolutely split locally free sheaves on Brauer–Severi varieties of index two

Let X be a Brauer–Severi variety over a field k associated with a central simple k-algebra of index two. This variety has the property of being isomorphic to a projective space after base change to a degree two Galois extension L. A locally free sheaf on X is called absolutely split if it splits after base change as a direct sum of invertible sheaves on the projective space.We classify the isomorphism classes of absolutely split locally free sheaves on X.