In https://arxiv.org/abs/2305.19777, there is a fact stated as folklore. Here it is.
Fact 2.7 (Folklore). For any $q\in\mathbb{N}$, lattice $\mathcal{L}\in\mathbb{R}^n$ of rank $k$ there exist linear independent vectors $u_1,\dots,u_k\in\mathcal{L}$ such that $\|u_i\|_q = \lambda_i^{(q)}(\mathcal{L})$.
In the statement, the $\lambda_i^{(q)}(\mathcal{L})$ are the successive minima defined with the $q$-norm.
This seems to be common knowledge. But as a novice in the field, I don't see how it follows from the definition of the successive minima. Can someone point me to some references that proves this common knowledge?