It is enough to consider \(J,K \in I_n\), since the signs factor out by bilinearity. For fixed \(r\), the vector \(Ae_r\) has coefficient \(2\) in coordinate \(r\), coefficient \(-1\) in the adjacent coordinates \(r-1\) and \(r+1\), and coefficient \(0\) elsewhere. Hence, for \(u\in [1,n]\),

where \(\delta _{x,y}\) is the Kronecker delta. Summing over \(u=i,\ldots ,j\) gives the displayed formula.

Write \(J=\varepsilon [i,j]\). In Lemma 3, the two positive endpoint terms are both \(1\), while the two negative endpoint conditions \(i=j+1\) are impossible because \(i\le j\). Thus

Write \(J=\varepsilon [i,j]\) and \(K=\delta [k,l]\). Let

By Lemma 3, \(\mathcal{L}(\alpha _K,\alpha _J^\vee )=\varepsilon \delta E\). The two negative endpoint conditions cannot both hold: if \(i=l+1\) and \(k=j+1\), then \(l{\lt}i\le j{\lt}k\le l\), a contradiction. Hence \(E\ge -1\). Since also \(E\le 2\), the equality \(\varepsilon \delta E=2\) forces \(\varepsilon \delta =1\) and \(E=2\).

The equality \(E=2\) forces both positive endpoint terms to be \(1\), so \(i=k\) and \(j=l\). Since \(\varepsilon \delta =1\) and \(\varepsilon ,\delta \in \{ \pm 1\} \), we also have \(\varepsilon =\delta \). Thus \(K=J\). The converse is Lemma 4.

Suppose first that \(\alpha _J=\alpha _K\). Pairing both sides with \(\alpha _J^\vee \) gives

Lemma 5 gives \(K=J\).

Now suppose that \(\alpha _J^\vee =\alpha _K^\vee \). Pairing both sides with \(\alpha _J\) gives

Applying Lemma 5 with the roles of the two signed intervals exchanged gives \(J=K\).

A direct case check from (??) shows that \(s_J(s_J(K))=K\) for every signed interval \(K\). Hence \(s_J\) is its own inverse, and therefore a permutation.

First assume that \(J=[i,j]\) and \(K=[k,l]\) are positive intervals. Put \(c=\mathcal{L}(\alpha _K,\alpha _J^\vee )\). Lemma 3 gives

Comparing this value with (??) gives:

Each entry in the last column is precisely \(\alpha _{s_J(K)}\).

For arbitrary signs, write \(J=\varepsilon J_0\) and \(K=\delta K_0\) with \(J_0,K_0\in I_n\) and \(\varepsilon ,\delta \in \{ \pm 1\} \). Since \(s_{\varepsilon J_0}(\delta K_0)=\delta s_{J_0}(K_0)\) and

the positive case, multiplied by \(\delta \), proves the signed case.

For positive intervals, symmetry of the type \(A\) Cartan matrix gives the same endpoint value for \(\mathcal{L}(\alpha _J,\alpha _K^\vee )\) as in the proof of Lemma 8. The case table there used only additivity of sums over adjacent or nested intervals. Replacing each \(Ae_r\) by \(e_r^*\) in that table gives the desired coroot identity for positive intervals. The signed case follows by the same sign reduction as in Lemma 8.

The quadruple \((\mathbb {Z},\mathbb {Z}^n,(\mathbb {Z}^n)^*,\mathcal{L})\) is the standard perfect pairing between a finite free \(\mathbb {Z}\)-module and its dual. It remains to verify the root-pairing axioms from Definition 1.

The maps \(\alpha \) and \(\alpha ^\vee \) are injective by Lemma 6. The identity \(\mathcal{L}(\alpha _J,\alpha _J^\vee )=2\) is Lemma 4. Lemma 7 gives \(s_J\in \operatorname {Perm}(I)\) for each \(J\). Finally, Lemmas 8 and 9 give the two reflection formulas. These are exactly the conditions in Definition 1. This proves the tuple is a root pairing. It is finite because \(I\) is finite, crystallographic by Lemma 3, and reduced by (??) and (??).