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What is the maximal number of orthogonal operators $ A _{1} , \dotsc, A _ {m} $ in $ \mathbb{Q}^{ n } $ satisfying the relations $ A_{i}^{2} = - I $ and $ A_{i}A_{j} + A_{j}A_{i} = 0 $ for $ i \neq j $ ? We can consider $ n= 6 $ .

If we take $ \mathbb{R} ^{n} $ instead of $ \mathbb{Q} ^{n} $ then the answer of the Hurwitz–Radon problem is $ \rho(n) - 1 $ where if $ n = (2a + 1 ) 2^{c+4d} $ where $ 0 \leq c \leq 3 $ then $ \rho(n) = 2^{c} + 8d $.

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    $\begingroup$ Wikipedia en.wikipedia.org/wiki/Hurwitz_problem gives a statement of a Hurwitz-Radon problem with no mention of operators, having to do with expressing a product of two sums of squares as a sum of squares. It has the same value of $\rho(n)$; is it the same problem? Wikipedia suggests the answer is independent of the field, so long as the characteristic isn't $2$. $\endgroup$
    – Gerry Myerson
    Commented May 16, 2021 at 0:42
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    $\begingroup$ Thanks @GerryMyerson , yah both definition are same , I have the followed proof from "Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022 " chapter 1 $\endgroup$
    – Sky
    Commented May 16, 2021 at 18:20

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