Geodesic
Clifford division algebras and anisotropic quadratic forms: two counterexamples
Glasgow mathematical journal, Tome 28 (1986) no. 2, pp. 227-228

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In a recent paper [3], D. W. Lewis proposed the following conjecture. (The notation is the same as that in [2] and [3].)Conjecture. Let F be a field of characteristic not 2 and let a1, b1..., an, bn ∈ Fx. The tensor product of quaternion algebrasis a division algebra if and only if the quadratic form over Fis anisotropic.This equivalence indeed holds for n = 1 as is well known [2, Theorem 2.7], and Albert [1] (see also [4, §15.7]) has shown that it also holds for n = 2. The aim of this note is to provide counterexamples to both of the implications for n ≥ 3. 
Mammone, P.; Tignol, J. P. Clifford division algebras and anisotropic quadratic forms: two counterexamples. Glasgow mathematical journal, Tome 28 (1986) no. 2, pp. 227-228. doi: 10.1017/S0017089500006558
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[1] 1.Albert, A. A., A construction of non-cyclic normal division algebras, Bull. Amer. Math. Soc. 38 (1932), 449–456. Google Scholar | DOI

[2] 2.Lam, T.-Y., The algebraic theory of quadratic forms (Benjamin, 1973). Google Scholar

[3] 3.Lewis, D. W., A note on Clifford algebras and central division algebras with involution, Glasgow Math. J. 26 (1985), 171–176. Google Scholar | DOI

[4] 4.Pierce, R. S., Associative algebras (Springer, 1982). Google Scholar | DOI

[5] 5.Tignol, J. P., Algèbres indècomposables d'exposant premier, to appear in Adv. in Maths. Google Scholar

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