Kummer Elements and the mod-3 Invariant of Albert Algebras

Let k be a field with characteristic not 2 and 3. Assume that k contains the cube roots of unity. Let J be a Titsʹ-first-construction Albert division algebra over k. In this paper we relate Kummer elements in J with the mod-3 invariant g3(J). We prove that if x J is a Kummer element with x3 = λ, then J J(D, λ) for some D, a degree-3 central division algebra over k. We show that if J1 = J(A, μ) and J2 = J(B, ν) are Titsʹ first-construction Albert division algebras with g3(J1) = g3(J2) then J2 J(D, μ) for some degree-3 central division algebra D over k.