The length of the shortest, nonzero vector in a lattice $\Gamma$ is denoted by $\lambda(\Gamma)$. The Hermite constant is then \begin{equation*} \gamma_d = \max_{L \ \text{d-rank lattice}} \frac {\lambda(L)^2} {|\mathrm{vol}(B)^{2/d}|} \end{equation*} It is known that $\gamma_d^d$ is always rational. In [LLL10, page 34] it is claimed that this is "because there is always an integral critical lattice" (a lattice is critical, if the maximum above is reached). I believe this not true (e.g. in 2 dimensions, the hexagonal lattice is the unique critical lattice but is not integral), instead it should be "there is always an integral critical quadratic form" (which correspond to lattices up to squaring).
However, I do not see why this is the case (if my claims above are indeed true). So why is there always a rational critical quadratic form, or equivalently always a critical lattice with coefficients whose squares are all rational?
[LLL10] Phong Q. Nguyen and Brigitte Vallée. The LLL Algorithm. Springer, 2010. https://www.ionica.nl/wp-content/uploads/2019/04/the-LLL-Algorithms.pdf
Well, I have found a solution now. A theorem by Voronoi states that
Extreme means that the "Hermite Quotient" $\lambda(Q)/|\det(Q)|^{1/d}$ of a positive definite quadratic form $Q$ has a local minimum at $Q$. Furthermore, a positive definite quadratic form $Q$ is called perfect, if it is uniquely determined by its shortest vectors, i.e. if it is the unique solution of the system of linear equations \begin{equation*} v^TXv = \lambda(Q) \quad \text{for all $v \in \mathbb{Z}^d$ with $v^TQv = 1$} \end{equation*} Eutacticity seems not so relevant for the argument, so I do not include a definition here (there does not seem to be a nice and short one).
Clearly, a critical quadratic form must be extreme, and thus perfect. Hence, it is the unique solution of a system of linear equations where all coefficients are integers, and thus is rational.