Let $\Phi$ be a crystallographic root system in a Euclidean vector space $V$. Let $S\subseteq \Phi$ be some subset of roots for which $\mathrm{Span}_{\mathbb{R}}(S)=V$.

Question: How big can $[\mathrm{Span}_{\mathbb{Z}}(\Phi):\mathrm{Span}_{\mathbb{Z}}(S)]$ be?

If $\Phi$ is of Type A, then I believe we always have $[\mathrm{Span}_{\mathbb{Z}}(\Phi):\mathrm{Span}_{\mathbb{Z}}(S)]=1$ because of total unimodularity.

But for instance, if $\phi=D_4$ and $S=\{\alpha_1,\alpha_1+2\alpha_2+\alpha_3+\alpha_4,\alpha_3,\alpha_4\}$, where $\alpha_2$ corresponds to the trivalent node, then I get $[\mathrm{Span}_{\mathbb{Z}}(\Phi):\mathrm{Span}_{\mathbb{Z}}(S)]=2$. (With the standard realization of $D_4$ this is $S=\{(1,-1,0,0),(1,1,0,0),(0,0,1,-1),(0,0,1,1)\}$.)

In particular I'd like to know if $[\mathrm{Span}_{\mathbb{Z}}(\Phi):\mathrm{Span}_{\mathbb{Z}}(S)]$ is absolutely bounded or not.

Let $\Phi$ be an irreducible crystallographic root system in a Euclidean vector space $V$. Let $S\subseteq \Phi$ be some subset of roots for which $\mathrm{Span}_{\mathbb{R}}(S)=V$.

Question: How big can $[\mathrm{Span}_{\mathbb{Z}}(\Phi):\mathrm{Span}_{\mathbb{Z}}(S)]$ be?

If $\Phi$ is of Type A, then I believe we always have $[\mathrm{Span}_{\mathbb{Z}}(\Phi):\mathrm{Span}_{\mathbb{Z}}(S)]=1$ because of total unimodularity.

But for instance, if $\phi=D_4$ and $S=\{\alpha_1,\alpha_1+2\alpha_2+\alpha_3+\alpha_4,\alpha_3,\alpha_4\}$, where $\alpha_2$ corresponds to the trivalent node, then I get $[\mathrm{Span}_{\mathbb{Z}}(\Phi):\mathrm{Span}_{\mathbb{Z}}(S)]=2$. (With the standard realization of $D_4$ this is $S=\{(1,-1,0,0),(1,1,0,0),(0,0,1,-1),(0,0,1,1)\}$.)

In particular I'd like to know if $[\mathrm{Span}_{\mathbb{Z}}(\Phi):\mathrm{Span}_{\mathbb{Z}}(S)]$ is absolutely bounded or not.