# Commentarii Mathematici Helvetici

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**Volume 74, Issue 2, 1999, pp. 297–305**

**DOI: 10.1007/s000140050090**

Isotopy and invariants of Albert algebras

^{[1]}(1) Library, Tata Institute of Fundamental Research, Homi Bhabha Road, 400 005, COLABA, MUMBAI, INDIA

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.

*Keywords: *Jordan algebras, isotopy, Albert algebras, invariants

. Isotopy and invariants of Albert algebras. *Comment. Math. Helv.* 74 (1999), 297-305. doi: 10.1007/s000140050090