Rank-two systems A_2, B_2 and G_2

For $A_2$ take ${\mathrm{\Phi }}_A=\{u_0,\dots ,u_5\}$. This finite set spans the plane, is reduced, and each root reflection is a symmetry of the regular hexagon. Its Cartan integers are $2\cos \theta \in \{0,\pm 1,\pm 2\}$. Put $\alpha =u_0$ and $\beta =u_2$; then ${\mathrm{\Phi }}_A^{+}=\{\alpha ,\beta ,\alpha +\beta \}$ is a positive system with base $(\alpha ,\beta )$, and $(\alpha ,\beta )=-1\mathrm{/}2$. Thus its Cartan matrix is $\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$ and its off-diagonal Cartan product is $1$.

For $B_2$ take ${\mathrm{\Phi }}_B=\{\pm e_1,\pm e_2,\pm e_1\pm e_2\}$. This finite set spans the plane, omits zero, and is reduced. The reflections in the coordinate roots change one sign, while those in $e_1\pm e_2$ interchange the coordinates with possible sign changes, so every root reflection preserves ${\mathrm{\Phi }}_B$; direct pairings give Cartan integers in $\{0,\pm 1,\pm 2\}$. The roots $\alpha =e_1-e_2$ and $\beta =e_2$ form a base, since the positive roots are $\alpha ,\beta ,\alpha +\beta ,\alpha +2\beta$. Moreover $(\alpha ,\alpha )=2$, $(\beta ,\beta )=1$, and $(\alpha ,\beta )=-1$, so the Cartan matrix for $(\alpha ,\beta )$ is $\left(\begin{array}{cc}2& -1\\ -2& 2\end{array}\right)$ and its off-diagonal Cartan product is $2$.

For $G_2$, choose $\alpha ,\beta$ with $(\alpha ,\alpha )=6$, $(\beta ,\beta )=2$, $(\alpha ,\beta )=-3$ and put ${\mathrm{\Phi }}_G=\{\pm \alpha ,\pm \beta ,\pm (\alpha +\beta ),\pm (\alpha +2\beta ),\pm (\alpha +3\beta ),\pm (2\alpha +3\beta )\}\mathrm{.}$ The Gram determinant is positive, so $\alpha ,\beta$ form a basis; the displayed coefficient pairs then show that ${\mathrm{\Phi }}_G$ is finite, spans the plane, omits zero, and is reduced. The reflection $s_{\alpha }$ interchanges $\beta$ with $\alpha +\beta$ and $\alpha +3\beta$ with $2\alpha +3\beta$, and fixes $\alpha +2\beta$; the reflection $s_{\beta }$ interchanges $\alpha$ with $\alpha +3\beta$ and $\alpha +\beta$ with $\alpha +2\beta$, and fixes $2\alpha +3\beta$. Together with the images of $\alpha$ and $\beta$, these permutations show that the long and short roots are the two orbits of the simple reflections. Conjugating $s_{\alpha }$ or $s_{\beta }$ therefore proves reflection closure for every root. Direct pairings give integral Cartan integers in $\{0,\pm 1,\pm 2,\pm 3\}$. The six displayed unnegated roots are positive and have base $(\alpha ,\beta )$, whose Cartan matrix is $\left(\begin{array}{cc}2& -1\\ -3& 2\end{array}\right)$ and whose off-diagonal Cartan product is $3$.

By [L1] the three systems are exactly the irreducible rank-two reduced crystallographic systems, and the displayed Cartan products $1,2,3$ are those of $A_2,B_2,G_2$ respectively.