Convert .car file PBC to basis vectors?
I found the way to convert the hexagonal parameters:

I have a,b,c and $\alpha$, $\beta$, $\gamma$. The solution in basis vectors is:

basis1 = (a, 0, 0)
basis2 = (xy, yx, 0)
basis3 = (xz, yz, c)


where
\begin{align}
xy &amp;= b\cdot\cos(\gamma)\\
yx &amp;= b\cdot\sin(\gamma)\\
xz &amp;= c\cdot\cos(\beta)\\
yz &amp;= \frac{b\cdot c\cdot \cos(\alpha) - xy \cdot xz}{\sqrt{b^2 - xy^2}}\\
\end{align}

Thus my solution was:

basis1 = (27.1979,  0,       0     )
basis2 = (-13.5990, 23.5541, 0     )
basis3 = (0,        0,       15.499)