Every Gaussian-integer icube in dimension 4, and every one in dimension 3 whose norm is a sum of two squares, extends to an equal-length integral orthogonal basis; in dimension 4k+2 the norm must be a sum of two squares.
On rational points of orthogonal group
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abstract
Let $n$ be a positive integer. We show that a unit rational space vector whose multiple by $n$ is an integer vector can be extended to a rational orthonormal basis whose all members have the same property.
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Extensions of integral orthoregular sets and icubes
Every Gaussian-integer icube in dimension 4, and every one in dimension 3 whose norm is a sum of two squares, extends to an equal-length integral orthogonal basis; in dimension 4k+2 the norm must be a sum of two squares.