Let $v_1\cdots v_6$ be six unit vectors in $3$D and $\theta_{ij}$ denote the angle between the vectors $v_i$ and $v_j$. These vectors will form, in total, $^6C_2=15$ angles between them. Suppose we know the exact value of three of them, $$\theta_{12}=\theta_{23}=\theta_{31}=109.5^{\circ}$$ Is there any way that I could represent the remaining twelve angles as some function of the following angles? $$\theta_{45},\theta_{56},\theta_{46},\theta_{41},\theta_{52},\theta_{63}$$
Why do I think it should be possible?
If you take a look at the angles listed above you'll see that each vector occurs exactly three times in total. For example we have the angles $\theta_{45},\theta_{46},\theta_{41}$ corresponding to the vector $v_4$. Now in $3$D space you just need three angles to specify a vector. Since every vector satisfies this condition, this should lead to a unique orientation of the six vectors, locking the remaining angles into place (or atleast I hope so).
I'm unsure where to even start with this honestly. My ultimate aim is to reduce the parameters as much as I can so that I can carry out an optimization.