In Wikipedia it is stated that the volume of the parallelepiped given its edge lengths $a,b,c$, and the internal angles between the edges $\alpha ,\beta ,\gamma $ is:
\begin{equation*} V=abc\sqrt{1+2\cos \alpha \cos \beta \cos \gamma -\cos ^{2}\alpha - \cos^{2}\beta - \cos ^{2}\gamma }\qquad(*). \end{equation*}
I was not able to derive it by using the determinant formula and expressing $\cos \alpha ,\cos \beta ,\cos \gamma $ in terms of $a,b,c$ and $\alpha ,\beta ,\gamma $. For instance
\begin{equation*} a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}=ab\cos \alpha . \end{equation*}
Question: Could you give a hint on how can the formula (*) be proved?