A parallelogram is constructed on vectors $\overline{a}=3\overline{\alpha }-\overline{\beta },\overline{b}=\overline{\alpha }+3\overline{\beta },if\left|\overline{\alpha }\right|=\left|\overline{\beta }\right|=2$ and angle between  $\overline{\alpha }and\overline{\beta }$ is $\frac{\pi }{3},$ then the length of a diagonal of parallelogram is 

$\overrightarrow{\mathrm{a}}\text{ and }\overrightarrow{\mathrm{b}}\text{ are two adjacent sides of }1\mathrm{lgm}\text{ so sum of sides (adjacent) will give length of diagonal. }$

$\begin{array}{r}a=3\alpha -\beta \text{ of }\alpha +3\beta \\ a+b=4\alpha +2\beta \\ |\overrightarrow{a}+\overrightarrow{b}|=\sqrt{(4\alpha {)}^2+(2\beta {)}^2+2\left(4\alpha \right)\cdot \left(2\beta \right)}\\ =\sqrt{16|\alpha {|}^2+4|\beta {|}^2+16\alpha \cdot \beta }\\ =\sqrt{16\left(4\right)+4\left(4\right)+16\left|\alpha \right|\left|\beta \right|\cos \theta }\end{array}$

$\begin{array}{r}|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=\sqrt{64+16+64\cos \frac{\pi }{3}}\\ =\sqrt{64+16+32}\\ =\sqrt{112}=4\sqrt{7}\end{array}$

$\left|BD\right|=|a-b|=4\sqrt{3}$