By lattice I mean a subgroup of the additive group $\Bbb R^n$ that has full rank (that is, it spans $\Bbb R^n$) and which is also isomorphic to $\Bbb Z^n$. A lattice $\mathcal L$ is integral if any $v,w\in\mathcal L$ have $\langle v,w\rangle\in\Bbb Z$, where $\langle\cdot,\cdot\rangle$ denotes the standard inner product.

I wonder whether every integral lattice $\mathcal L$ is a sublattice of $\Bbb Z^n$ in the following sense:

Consider some vectors $v_1,..., v_k\in\Bbb Z^n$. Then $$\mathcal L(v_1,...,v_k):=\mathrm{span}\{v_1,...,v_k\}\cap \Bbb Z^n$$ is an integral lattice in $\mathrm{span}\{v_1,...,v_k\}\subseteq\Bbb R^n$ (with the inner product inherited from $\Bbb Z^n$), and will be called a sublattice of $\Bbb Z^n$.

Being a sublattice, of course, is meant as being isomorphic to such one in some natural sense. For example, $\mathcal L$ is considered a sublattice of $\Bbb Z^n$ if there is an actual sublattice $\mathcal L(v_1,...,v_k)\subseteq\Bbb Z^n$, a group isomorphism $\phi:\mathcal L\to\mathcal L(v_1,...,v_k)$ and a constant $\alpha\in\Bbb R$, so that $$\langle \phi(v),\phi(w)\rangle=\alpha \langle v,w\rangle,\quad\text{for all $v,w\in\mathcal L$}.$$

If not all integral lattices are such sublattices, are there some natural restriction (e.g. unimodularity) to make this true?