{\displaystyle \langle \beta ,\alpha \rangle \langle \alpha ,\beta \rangle =2{\frac {(\alpha ,\beta )}{(\alpha ,\alpha )}}\cdot 2{\frac {(\alpha ,\beta )}{(\beta ,\beta )}}=4{\frac {(\alpha ,\beta )^{2}}{\vert \alpha \vert ^{2}\vert \beta \vert ^{2}}}=4\cos ^{2}(\theta )=(2\cos(\theta ))^{2}\in \mathbb {Z} .}