A root system of rank at most 16 with 480 roots, all of squared norm 2, is D16 or E8 plus E8

The hypothesis gives $⟨\gamma ,\gamma ⟩=2$ for every root $\gamma$, so for roots $\alpha ,\beta$ the Cartan integer $2⟨\beta ,\alpha ⟩\mathrm{/}⟨\alpha ,\alpha ⟩$ equals $⟨\beta ,\alpha ⟩\in \mathbb{Z}$ by [F1]. For nonparallel roots, $\alpha +\beta$ and $\alpha -\beta$ are nonzero, so positive definiteness and bilinearity give $0<⟨\alpha \pm \beta ,\alpha \pm \beta ⟩=4\pm 2⟨\beta ,\alpha ⟩$; hence $-2<⟨\beta ,\alpha ⟩<2$. This integer is therefore $-1$, $0$, or $1$. In any base, distinct simple roots are nonparallel and each has self-inner-product $2$, so their Cartan integers are both zero or both equal to the same $1$ or $-1$; their product is therefore $0$ or $1$, and the diagram has only single edges. By [F3], every irreducible component has type $A_n$, $D_n$, $E_6$, $E_7$, or $E_8$.

For $n\ge 1$, in $H_n:=\{x\in {\mathbb{R}}^{n+1}:{\sum }_{i=1}^{n+1}{x}_i=0\}$ define ${\mathrm{\Psi }}_{A_n}:=\{e_i-e_j:i\mathrm{\ne }j\}$, where $e_1,\dots ,e_{n+1}$ is the standard basis of ${\mathbb{R}}^{n+1}$. For $n\ge 4$, in ${\mathbb{R}}^n$ define ${\mathrm{\Psi }}_{D_n}:=\{\pm e_i\pm e_j:i<j\}$. In each set all roots have squared norm $2$, pairings are integers, and the only roots on the line through a root $\alpha$ are $\pm \alpha$. For ${\mathrm{\Psi }}_{A_n}$, reflection in $e_i-e_j$ exchanges coordinates $i,j$. For ${\mathrm{\Psi }}_{D_n}$, reflection in $e_i-e_j$ exchanges those coordinates and reflection in $e_i+e_j$ exchanges and negates them. These maps preserve the respective root sets, so both are reduced crystallographic root systems. The vectors ${\alpha }_i=e_i-e_{i+1}$ for $1\le i\le n$ form a basis of $H_n$ for ${\mathrm{\Psi }}_{A_n}$: in a zero linear combination, coordinates $1,\dots ,n$ successively force each coefficient to vanish. Every root $e_i-e_j$ with $i<j$ is ${\alpha }_i+\cdots +{\alpha }_{j-1}$. For ${\mathrm{\Psi }}_{D_n}$ take ${\alpha }_i=e_i-e_{i+1}$ for $1\le i<n$ and ${\alpha }_n=e_{n-1}+e_n$; coordinates $1,\dots ,n-2$ in a zero linear combination force the first $n-2$ coefficients to vanish, and coordinates $n-1,n$ then force the last two to vanish. These $n$ independent vectors form a basis of ${\mathbb{R}}^n$. For $i<j$, $e_i-e_j={\sum }_{k=i}^{j-1}{\alpha }_k$; for $i<j<n$, $e_i+e_j={\sum }_{k=i}^{j-1}{\alpha }_k+2{\sum }_{k=j}^{n-2}{\alpha }_k+{\alpha }_{n-1}+{\alpha }_n$; and for $i<n$, $e_i+e_n={\sum }_{k=i}^{n-2}{\alpha }_k+{\alpha }_n$, with empty sums taken as zero. Thus in both models every root has one sign of integral simple-root coordinates. Distinct basis roots in ${\mathrm{\Psi }}_{A_n}$ have inner product $-1$ for consecutive indices and $0$ otherwise. In ${\mathrm{\Psi }}_{D_n}$ the inner product is $-1$ for consecutive indices $1\le i\le n-3$ and for the pairs $(n-2,n-1)$ and $(n-2,n)$, and is $0$ for every other distinct pair. Thus the diagrams are a path for ${\mathrm{\Psi }}_{A_n}$ and the trivalent diagram with arms of lengths $1,1,n-3$ for ${\mathrm{\Psi }}_{D_n}$; both diagrams are connected. Because each simple-root basis is connected and spans its space, every root is nonorthogonal to at least one basis root, so the nonorthogonality graph is connected and both systems are irreducible by [F2]. By [F3], these coordinate systems realize the types $A_n$ and $D_n$ used in its classification; their root counts are therefore the counts of those types, since a root-system isomorphism is a bijection on root sets [F4]. The counts are $\mathrm{\mid }{A}_n\mathrm{\mid }=2\left(\genfrac{}{}{0px}{}{n+1}{2}\right)=n(n+1)$ and $\mathrm{\mid }{D}_n\mathrm{\mid }=4\left(\genfrac{}{}{0px}{}{n}{2}\right)=2n(n-1)$. The $D_n$ roots generate $D_n=\{z\in {\mathbb{Z}}^n:{\sum }_i{z}_i\in 2\mathbb{Z}\}$: for $z\in D_n$, the coefficients $c_i={\sum }_{j\le i}{z}_j$ for $i\le n-2$, $c_n=({\sum }_{j\le n}{z}_j)\mathrm{/}2$, and $c_{n-1}=c_n-z_n$ are integers and give $z={\sum }_{i=1}^{n-1}{c}_i(e_i-e_{i+1})+c_n(e_{n-1}+e_n)$.

For systems whose components have only types $A,D$, set $g(1)=2$, $g(2)=6$, $g(3)=12$, and $g(r)=2r(r-1)$ for $r\ge 4$. Step 1.2 shows that every component of rank $r$ has at most $g(r)$ roots: for $r\le 3$ the listed type is $A_r$ using the low-rank identifications in [F3], and for $r\ge 4$ the $A_r$ count $r(r+1)$ is at most the $D_r$ count $2r(r-1)$. The function $g$ is increasing and strictly superadditive. For $a,b\ge 4$, $g(a+b)-g(a)-g(b)=4ab>0$. By symmetry take $a\le b$; the small cases $a,b\le 3$ are $g(2)>2g(1)$, $g(3)>g(1)+g(2)$, $g(4)>g(1)+g(3)$, $g(4)>2g(2)$, $g(5)>g(2)+g(3)$, and $g(6)>2g(3)$. For $a=1,2,3$ and $b\ge 4$, the differences are respectively $4b-2$, $8b-2$, and $12b$, all positive. Consequently, components of total rank $s$ with types only $A,D$ have at most $g(s)$ roots, with strict inequality if there is more than one component.

In ${\mathrm{\Phi }}_8$ the integer-coordinate roots are precisely $\pm e_i\pm e_j$ and number $4\left(\genfrac{}{}{0px}{}{8}{2}\right)=112$. A half-integer vector has norm squared at least $8\mathrm{/}4=2$, with equality exactly when every coordinate is $\pm 1\mathrm{/}2$; the sum-even condition permits exactly the sign patterns with an even number of minus signs, hence $2^7=128$ roots. Thus $\mathrm{\mid }{\mathrm{\Phi }}_8\mathrm{\mid }=240$. The roots $e_i-e_{i+1}$ for $1\le i\le 7$ and $e_7+e_8$ are linearly independent: in a zero linear combination, coordinates $1$ through $6$ successively force the first six coefficients to vanish, and coordinates $7,8$ then force the last two to vanish. Thus they form a basis of ${\mathbb{R}}^8$. Because $E_8$ is integral by [F5], for norm-2 roots $\alpha ,\beta$ the integer $⟨\beta ,\alpha ⟩$ is integral and reflection sends $\beta$ to $\beta -⟨\beta ,\alpha ⟩\alpha \in E_8$ while preserving norm; the resulting vector is again in ${\mathrm{\Phi }}_8$. All roots are nonzero and have the same norm, so the set is reduced. It is finite by the coordinate count, hence is a reduced crystallographic root system. The eight displayed basis roots have a connected nonorthogonality graph: consecutive differences pair to $-1$, and $e_7+e_8$ pairs to $-1$ with $e_6-e_7$. Every root is nonorthogonal to at least one of these basis roots, since they span ${\mathbb{R}}^8$ and every root is nonzero. Therefore the full nonorthogonality graph is connected, and [F2] makes ${\mathrm{\Phi }}_8$ irreducible. Its rank is $8$; the irreducible types of rank $8$ with only single-edge diagrams other than $E_8$ are $A_8$ and $D_8$, whose root counts $72$ and $112$ from step 1.2 are smaller than $240$, so [F3] identifies ${\mathrm{\Phi }}_8$ as type $E_8$. Finally $D_8\subseteq Q({\mathrm{\Phi }}_8)$ and $h_8\in {\mathrm{\Phi }}_8$, so $Q({\mathrm{\Phi }}_8)=D_8\cup (D_8+h_8)=E_8$.

The seven roots $u_1=e_1-e_2,u_2=e_2-e_3,u_3=e_3-e_4,u_4=e_4-e_5,u_5=e_5-e_6,u_6=e_7-e_8$ and $v:=(-e_1+e_2+e_3+e_4+e_5+e_6+e_7-e_8)\mathrm{/}2$ lie in ${\mathrm{\Phi }}_7$. They are independent: summing coordinates $1$ through $6$ in a zero linear relation kills all $u_i$ terms and gives $2$ times the coefficient of $v$, so that coefficient is zero; coordinates $1$ through $5$ then successively force the coefficients of $u_1,\dots ,u_5$ to vanish, and coordinate $7$ forces the coefficient of $u_6$ to vanish. The six roots $w_1=e_1-e_2,w_2=e_2-e_3,w_3=e_3-e_4,w_4=e_4-e_5,w_5=e_1+e_2$ and $v$ lie in ${\mathrm{\Phi }}_6$. In a zero linear relation, coordinate $6$ forces the coefficient of $v$ to vanish; then $w_1,\dots ,w_4$ are independent and span the sum-zero subspace of the first five coordinates, while $w_5$ has coordinate sum $2$, so all five remaining coefficients vanish. These independent sets span $W_7$ and $W_6$, respectively. Each ${\mathrm{\Phi }}_j$ is finite, consists of nonzero equal-length roots, inherits integrality and reducedness from ${\mathrm{\Phi }}_8$, and spans $W_j$. If $\alpha ,\beta \in {\mathrm{\Phi }}_j$, reflection $s_{\alpha }\beta$ remains in ${\mathrm{\Phi }}_8$ by step 2.2 and in $W_j$ because $\alpha$ is orthogonal to every normal vector defining $W_j$; hence each is a root system. Counting in the $112$ integer and $128$ half-integer roots gives $\mathrm{\mid }{\mathrm{\Phi }}_7\mathrm{\mid }=62+64=126$ (integer roots are supported in the first six coordinates or on the pair $7,8$ with opposite signs; half-roots have opposite signs in positions $7,8$ and an odd number of minus signs among the first six) and $\mathrm{\mid }{\mathrm{\Phi }}_6\mathrm{\mid }=40+32=72$ (integer roots are supported in the first five coordinates; half-roots have $x_6=x_7=-x_8$ and the first-five sign parity is then forced).

The rank-6 system ${\mathrm{\Phi }}_6$ cannot have only $A$ or $D$ components, because step 2.1 bounds these by $g(6)=60$ roots; it has no component of rank greater than $6$, so its $72$ roots force one irreducible $E_6$ component occupying all six ranks. If the rank-7 system ${\mathrm{\Phi }}_7$ had no $E_7$ component, it would have at most $g(7)=84$ roots from $A,D$ components or $72+g(1)=74$ roots if it contained $E_6$, both below $126$; hence it is type $E_7$. Thus the coordinate root counts of types $E_6,E_7,E_8$ are $72,126,240$.

By [F2], $\mathrm{\Phi }$ is the orthogonal union of its irreducible components, each with a diagram having only single edges. If all components have type $A$ or $D$, step 2.1 gives at most $g(16)=480$ roots, with equality only for a single $D_{16}$ component. If exactly one exceptional component occurs, the maximum totals including all remaining $A,D$ components are $72+g(10)=252$ for $E_6$, $126+g(9)=270$ for $E_7$, and $240+g(8)=352$ for $E_8$. With two exceptional components the maximal possibilities are $E_6+E_6$ with $144+g(4)=168$, $E_6+E_7$ with $198+g(3)=210$, $E_6+E_8$ with $312+g(2)=318$, $E_7+E_7$ with $252+g(2)=258$, $E_7+E_8$ with $366+g(1)=368$, and $E_8+E_8$ with $480$ and no remaining rank; three exceptional components exceed rank $16$. Thus $480$ roots occur only for $D_{16}$ or $E_8\oplus E_8$, and both have rank $16$.

In the $D_{16}$ case, the standard roots $\pm e_i\pm e_j$ generate the even-coordinate-sum lattice $D_{16}$ by step 1.2. In the $E_8\oplus E_8$ case, the orthogonal root components each generate $E_8$ by step 2.2, so their root lattice is $E_8\oplus E_8$. A root-system isomorphism preserves integer spans and, because all roots have norm squared $2$, preserves their inner products; therefore the root lattice $Q(\mathrm{\Phi })$ is isometric to the corresponding standard coordinate lattice. The two root-system types are distinct because $D_{16}$ is irreducible while $E_8\oplus E_8$ has two irreducible components.