I'm having issues converting from a non-orthogonal coordinate space (abc) to a cartesian space (xyz).
I think I may have just thought myself into a corner, and I can't get out.
The systems have a common origin. $\mathbf{a}$ is parallel to $\mathbf{x}$. The $ab$ plane is in the $xy$ plane. The angle between $\mathbf{a}$ and $\mathbf{b}$ is $\gamma$, $\mathbf{b}$ and $\mathbf{c}$ is $\alpha$, and $\mathbf{a}$ and $\mathbf{c}$ is $\beta$. All angles are less than $\frac{\pi}{2}$ and greater than $\frac{\pi}{3}$.
So far, I've figured out that my transformation matrix is
$$ \begin{bmatrix} 1 & \cos\gamma & ?\\ 0 & \sin\gamma & ?\\ 0 & 0 & \sin\alpha\sin\beta \end{bmatrix} $$
Any light you can shed? Everything else that I've seen has been limited to 2D.
Edit: Now I'm not too sure on the last entry in that matrix...
So we need to find the coordinates of the unit vectors of $abc$ space in terms of $xyz$ space. You have found the first two. Let the $xyz$ coordinates of the unit vector in $c$ direction be $(x,y,z)$. It satisfies:
$$(1,0,0)\cdot(x,y,z)=\cos{\beta}\\ (\cos{\gamma},\sin{\gamma},0)\cdot(x,y,z)=\cos{\alpha}\\ x^2+y^2+z^2=1$$
where I used the fact that $\vec{u}\cdot\vec{v}=||\vec{u}|| ||\vec{v}||\cos{\theta}$. They can be easily solved. The $(x,y,z)$ will make the 3rd column.