Existence of each classified root system

(Type $A_n$.) In $E=\{x\in {\mathbb{R}}^{n+1}:{\sum }_i{x}_i=0\}$ put $\mathrm{\Phi }=\{e_i-e_j:i\mathrm{\ne }j\}$; it is finite, nonempty, spans $E$, and contains no zero vector. For $\alpha =e_i-e_j$ the reflection $s_{\alpha }$ sends $e_i$ to $e_j$, $e_j$ to $e_i$ and fixes every other coordinate vector, so it permutes $\mathrm{\Phi }$ and $\mathrm{\Phi }\cap \mathbb{R}\alpha =\{\pm \alpha \}$; the Cartan integers are $2(e_k-e_l,e_i-e_j)\mathrm{/}2\in \{0,\pm 1,\pm 2\}$. For the regular functional $x\mapsto (x,(n+1,n,\dots ,1))$, the positive roots are $e_i-e_j$ with $i<j$; each is ${\sum }_{k=i}^{j-1}(e_k-e_{k+1})$, and only the adjacent differences are indecomposable. Thus the simple roots are $e_i-e_{i+1}$ for $1\le i\le n$, whose Cartan matrix is $A_n$.

(Types $B_n$ and $C_n$.) In ${\mathbb{R}}^n$ put ${\mathrm{\Phi }}_B=\{\pm e_i\}\cup \{\pm e_i\pm e_j:i<j\}$ and ${\mathrm{\Phi }}_C=\{\pm 2e_i\}\cup \{\pm e_i\pm e_j:i<j\}$. Both are finite, span ${\mathbb{R}}^n$, omit $0$, and are reduced. Reflections: $s_{e_i}$ negates the $i$-th coordinate, $s_{2e_i}$ does the same, and $s_{e_i\pm e_j}$ permutes or changes the signs of coordinates $i,j$ and fixes the others, so each reflection permutes ${\mathrm{\Phi }}_B$ and ${\mathrm{\Phi }}_C$. Integrality is checked directly from $(e_k,e_l)={\delta }_{kl}$: for ${\mathrm{\Phi }}_B$ the Cartan integers lie in $\{0,\pm 1,\pm 2\}$, and for ${\mathrm{\Phi }}_C$ the values $2(\beta ,2e_i)\mathrm{/}(2e_i,2e_i)={\beta }_i$ and $2(\beta ,\alpha )\mathrm{/}(\alpha ,\alpha )$ with $\alpha$ of the second kind are likewise integers in $\{0,\pm 1,\pm 2\}$. For a regular vector with $t_1>\cdots >t_n>0$, direct expansion of the positive roots $e_i-e_j,e_i+e_j,e_i$ for $B_n$, and $e_i-e_j,e_i+e_j,2e_i$ for $C_n$, shows that their simple roots are respectively $e_1-e_2,\dots ,e_{n-1}-e_n,e_n$ and $e_1-e_2,\dots ,e_{n-1}-e_n,2e_n$; their Cartan matrices are $B_n$ and $C_n$.

(Type $D_n$.) In ${\mathbb{R}}^n$ put ${\mathrm{\Phi }}_D=\{\pm e_i\pm e_j:i<j\}$ for $n\ge 4$. It is finite, spans, is reduced, and each reflection $s_{e_i\pm e_j}$ fixes the other coordinates or changes their signs, so it permutes ${\mathrm{\Phi }}_D$; the Cartan integers are $0,\pm 1,\pm 2$. For a regular vector with $t_1>\cdots >t_n>0$, direct expansion of the positive roots $e_i-e_j,e_i+e_j$ shows that the simple roots are $e_1-e_2,\dots ,e_{n-2}-e_{n-1},e_{n-1}-e_n,e_{n-1}+e_n$; their Cartan matrix is $D_n$.

(Type $G_2$.) In ${\mathbb{R}}^2$ let $\alpha ,\beta$ satisfy $(\alpha ,\alpha )=6$, $(\beta ,\beta )=2$, $(\alpha ,\beta )=-3$ and put ${\mathrm{\Phi }}_G=\{\pm \alpha ,\pm \beta ,\pm (\alpha +\beta ),\pm (\alpha +2\beta ),\pm (\alpha +3\beta ),\pm (2\alpha +3\beta )\}$; the twelve vectors are distinct and nonzero. Direct computation gives $(\alpha ,\alpha +\beta )=3$, $(\alpha ,\alpha +2\beta )=0$, $(\alpha ,\alpha +3\beta )=-3$, $(\alpha ,2\alpha +3\beta )=3$, $(\beta ,\alpha +\beta )=-1$, $(\beta ,\alpha +2\beta )=1$, $(\beta ,\alpha +3\beta )=3$, $(\beta ,2\alpha +3\beta )=0$, from which the Cartan integers with denominator root $\alpha$ or $\beta$ are seen to lie in $\{0,\pm 1,\pm 2,\pm 3\}$ and every root is reduced; the same formulae show $s_{\alpha }$ permutes ${\mathrm{\Phi }}_G$ (it sends $\beta \mapsto \alpha +\beta$, $\alpha +\beta \mapsto \beta$, $\alpha +3\beta \mapsto 2\alpha +3\beta$, $2\alpha +3\beta \mapsto \alpha +3\beta$, and fixes $\alpha +2\beta$) and $s_{\beta }$ permutes ${\mathrm{\Phi }}_G$ (it sends $\alpha \mapsto \alpha +3\beta$, $\alpha +\beta \mapsto \alpha +2\beta$, $\alpha +2\beta \mapsto \alpha +\beta$, $\alpha +3\beta \mapsto \alpha$, and fixes $2\alpha +3\beta$). These permutations put every root in the orbit of $\alpha$ or $\beta$: the long positive roots are $\alpha ,\alpha +3\beta ,2\alpha +3\beta$ and the short ones are $\beta ,\alpha +\beta ,\alpha +2\beta$; their negatives are reached by the corresponding simple reflection and conjugation. Since these permutations are orthogonal, $s_{w\delta }=w{s}_{\delta }{w}^{-1}$ proves reflection invariance for every root, and invariance of inner products reduces all Cartan integers to the two denominator roots already checked. Choosing a regular vector positive on the six displayed unnegated roots makes $\alpha ,\beta$ the only indecomposable positive roots, since the other four roots have the decompositions $\alpha +\beta$, $(\alpha +\beta )+\beta$, $(\alpha +2\beta )+\beta$, and $\alpha +(\alpha +3\beta )$ into two positive roots. The coefficient pairs show that $\alpha ,\beta$ cannot so decompose. Their Cartan matrix is $\left(\begin{array}{cc}2& -1\\ -3& 2\end{array}\right)$, which is the $G_2$ diagram.

(Type $F_4$.) In ${\mathbb{R}}^4$ put ${\mathrm{\Phi }}_F=\{\pm e_i\}\cup \{\pm e_i\pm e_j:i<j\}\cup \{\frac{1}{2}{\sum }_{i=1}^4{ϵ}_i{e}_i:{ϵ}_i=\pm 1\}$. It has $8+24+16=48$ roots, spans, and is reduced by inspection of the coordinate supports and absolute values. Reflections in coordinate roots and in two-coordinate roots are signed coordinate permutations, hence preserve the set. For a half-root $h=\frac{1}{2}\sum {ϵ}_i{e}_i$, of squared norm $1$, use $s_h(x)=x-2(x,h)h$. A coordinate root maps to a half-root. For $x=u{e}_i+v{e}_j$, $u,v\in \{\pm 1\}$, the inner product is $0$ or $\pm 1$; in the latter case the reflection cancels the two occupied coordinates and leaves coefficients $\pm 1$ in the other two coordinates. For another half-root $k$, let $m$ be the number of agreeing signs. Then $(h,k)=(m-2)\mathrm{/}2$. If $m=0,4$ the roots are opposite or equal and reflection negates $k$; if $m=2$ it fixes $k$; if $m=1,3$ it gives a coordinate root. Thus all reflections preserve ${\mathrm{\Phi }}_F$. For a denominator root of norm $1$, all inner products are half-integers, so its Cartan integers are integral. For a denominator root of norm $2$ (a two-coordinate root), all inner products are integers, including those with half-roots; this checks the other denominators. The coordinate roots all lie in one part of any orthogonal partition, since $e_i+e_j$ connects $e_i$ to $e_j$. Every other root pairs nontrivially with a coordinate root. Hence the system is irreducible. By [L2], rank four permits $A_4,B_4,C_4,D_4,F_4$; the first four constructions have respectively $20,32,32,24$ roots. Thus the 48-root system has diagram $F_4$.

(Type $E_8$.) Put ${\mathrm{\Phi }}_8=\{\pm e_i\pm e_j:i<j\}\cup \{\frac{1}{2}{\sum }_{i=1}^8{ϵ}_i{e}_i:{ϵ}_i=\pm 1,{\prod }_i{ϵ}_i=1\}$. All roots have squared norm $2$; the $D_8$ subset spans, and reducedness is immediate. Reflections in its integer roots permute coordinates and change either zero or two signs, preserving both subsets. For two half-roots $h,k$, the number $m$ of agreeing signs is even and $(h,k)=(m-4)\mathrm{/}2$. For $m=0,8$ reflection negates $k$; for $m=4$ it fixes $k$; for $m=2,6$ the vector $k-(k,h)h$ has exactly two nonzero entries, both $\pm 1$, hence is an integer root. For a half-root $h$ and integer root $x=u{e}_i+v{e}_j$, $(x,h)$ is $0$ or $\pm 1$. If it is zero reflection fixes $x$. Otherwise $x-(x,h)h$ is a half-root: its signs differ from those of $h$ in six positions when $(x,h)=1$, and in two positions when $(x,h)=-1$, so the parity remains even. All pairwise inner products are integers by these formulas and the integer-root calculation; with norm squared $2$ this is crystallographic integrality. The integer roots form an irreducible spanning subset: their displayed $D_8$ base has a connected chain with a fork, so all its members lie in one part of any orthogonal partition; spanning then excludes a root in the other part. Thus ${\mathrm{\Phi }}_8$ is irreducible. It has $4\left(\genfrac{}{}{0px}{}{8}{2}\right)+2^7=240$ roots. Equal root lengths force every diagram edge to be simple by [L1], so [L2] leaves $A_8,D_8,E_8$. The first two have $72$ and $112$ roots by their explicit constructions; hence the diagram is $E_8$.

(The two restrictions.) Let $V_7=(e_7+e_8{)}^{\perp }$, $V_6=V_7\cap (e_6+e_8{)}^{\perp }$, and ${\mathrm{\Phi }}_i={\mathrm{\Phi }}_8\cap V_i$. Reflection in a root of $V_i$ preserves $V_i$, so reflection closure, integrality and reducedness are inherited. In ${\mathrm{\Phi }}_7$ the integer roots are the $60$ roots $\pm e_i\pm e_j$ on the first six coordinates and $\pm (e_7-e_8)$. Half-roots have ${ϵ}_7=-{ϵ}_8$, giving two choices for the last pair and $32$ odd-parity choices on the first six coordinates: $64$ half-roots, thus $126$ roots in all. The integer roots span $V_7$. In ${\mathrm{\Phi }}_6$, the last three coordinates obey $x_6=x_7=-x_8$. The integer roots are precisely the $40$ two-coordinate roots on the first five coordinates. The half-roots have last signs $(+,+,-)$ or $(-,-,+)$, with respectively odd or even parity among the first five signs: $16+16=32$ choices. Thus there are $72$ roots. The integer roots span the first five coordinate directions, and any half-root adds the remaining direction of $V_6$, proving spanning and rank six.

(Irreducibility and identification of the restrictions.) The $D_6$ integer roots on the first six coordinates of ${\mathrm{\Phi }}_7$ cannot split between orthogonal parts: their connected standard base spans those six coordinates. Every half-root has nonzero projection on that span, so pairs nontrivially with at least one such root and lies in the same part. Each of $\pm (e_7-e_8)$ pairs nontrivially with each half-root. Thus all roots lie in one part. The same argument for ${\mathrm{\Phi }}_6$ uses the connected spanning $D_5$ base on the first five coordinates, and the nonzero projection there of every half-root. Both restrictions are therefore irreducible. Their roots have equal length, so their diagrams are simply laced. In rank six [L2] leaves $A_6,D_6,E_6$; the first two have $42,60$ roots, whereas ${\mathrm{\Phi }}_6$ has $72$, giving $E_6$. In rank seven the possibilities are $A_7,D_7,E_7$, with the first two counts $56,84$, whereas ${\mathrm{\Phi }}_7$ has $126$, giving $E_7$.

These constructions realize every positive-rank type stated, with the classical and $G_2$ diagrams computed from simple roots and the exceptional diagrams identified using the independently established classification and explicit root counts. The additional empty rank-zero system allowed by the local convention is realized in $E=0$, where the spanning and reflection axioms hold vacuously. No unsolved source exercise is used as a proof premise.