Isotopy and invariants of Albert algebras

Let k be a field with characteristic different from 2 and 3. Let B be a central simple algebra of degree 3 over a quadratic extension K/k, which admits involutions of second kind. In this paper, we prove that if the Albert algebras \( J(B,\sigma,u,\mu) \) and \( J(B,\tau,v,\nu) \) have same \( f_3 \) and \( g_3 \) invariants, then they are isotopic. We prove that for a given Albert algebra J, there exists an Albert algebra J' with \( f_3(J')=0 \), \( f_5(J')=0 \) and \( g_3(J')=g_3(J) \). We conclude with a construction of Albert division algebras, which are pure second Tits' constructions.