TITLE: Lattice reduction in R^3 (R^4) or what is fundamental domain for SL(3,Z) , (SL(4,Z)) ?
QUESTION [5 upvotes]: Consider a lattice in R^3.
Is the some "canonical" way or ways to choose basis in it ?
I mean in R^2 we can choose a basis |h_1| < |h_2| and |(h_2, h_1)| < 1/2 |h_1|. 
Considering lattices with fixed determinant and up to unitary transformations we get standard picture of the PSL(2,Z) acting on the upper half plane, which has a fundamental domain Im (tau)>1 Re(tau) <1/2. 
What are the similar results for other small dimensions R^3, R^4, C^4, C^8 ?
What are the algorithms to find such a lattice reductions ?

REPLY [7 votes]: The book by Terras "Harmonic analysis on symmetric spaces and applications" volume 2 has some stereoscopic pictures of the fundamental domains for some similar groups.