9.45 (1984)

Let $a$ be a vector in $\mathbb{R}^n$ with rational coordinates and set
$$S(a) = \{ka + b \mid k \in \mathbb{Z}, \ b \in \mathbb{Z}^n\}.$$ It is obvious that $S(a)$ is a discrete subgroup in $\mathbb{R}^n$ of rank $n$. Find necessary and sufficient conditions in terms of the coordinates of $a$ for $S(a)$ to have an orthogonal basis with respect to the standard scalar product in $\mathbb{R}^n$.