Generating vectors of the face-centered cubic lattice I can't find a reference for a set of generating vectors for the Tetrahedral-octahedral honeycomb lattice. I would like to know the "canonical" set and if possible a more general set described by angles (I think it would take three angles to completely describe it but I'm not sure on this either.)
 A: Here you go. A3 or D3, called face centered cubic
http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A3.html 
Note that an extra dimension is required to avoid square roots in the basis coordinates. So, the three basis vectors are in $\mathbb R^4.$
$$  \langle 1,-1,0,0 \rangle  $$
$$  \langle 0,1,-1,0 \rangle $$
$$ \langle 0,0, 1,-1 \rangle $$
EEEEEEEDDDDDDIIIIIIIITTTTTTTTTT: Here is a somewhat cleaner answer, after I checked it in Schiemann's reduction. The lattice you want is the one I asked about, all integer points in $\mathbb R^3$ such that 
$$ x+y+z \; \; \; \;  \mbox{is even.}  $$
A basis can be given in the same dimension by
$$  \langle 0,1,1 \rangle  $$
$$  \langle 1,0,1 \rangle $$
$$ \langle 1,1, 0 \rangle $$
Notice that the squared lengths are 2, and the inner products are 1. So the matrix of inner products, the Gram matrix is
$$   
 \left(  \begin{array}{rrr}
  2  &  1  &  1 \\
  1   &  2  &  1 \\
  1  &  1   &  2  
\end{array} 
  \right)  
  $$
The twelve lattice points nearest the origin, in this lattice, are
$$ (0,1,1), (1,0,1), (1,1,0),  $$
$$ (0,-1,1), (-1,0,1), (-1,1,0),  $$
$$ (0,1,-1), (1,0,-1), (1,-1,0),  $$
$$ (0,-1,-1), (-1,0,-1), (-1,-1,0)  $$
for kissing number 12, as they are all the same distance from the origin, $\sqrt 2.$
