Abstract: Over many fields, including finite fields and imaginary number fields, any central simple algebra of exponent 2 can be decomposed as a product of quaternion algebras. However, over arbitrary fields there exist examples of indecomposable algebras of exponent 2. For division algebras, indecomposability can be detected by non-trivial torsion in the Chow group of the associated Severi-Brauer variety. In this talk, we consider the analogous problem of decomposability for algebras with involution. We replace the Severi-Brauer variety with G/B, the variety of Borel subgroups of an algebraic group G of inner type Dn, and estimate the Chow group of G/B via the gamma-filtration on its Grothendieck group. In particular, we look at groups of type D4 and their associated trialitarian triples.