Lattice reduction in R^3 (R^4) or what is fundamental domain for SL(3,Z) , (SL(4,Z)) ? 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 ?
 A: In higher dimensions, there doesn't seem to be anything as nice as in two dimensions: the fundamental domains get substantially more complicated and the algorithms become much less efficient.  However, there are still some beautiful results.  For example, Minkowski reduction is a natural generalization of the two-dimensional case.  See, for example, Chapter 2 of Computational geometry of positive definite quadratic forms by Achill Schürmann.  Minkowski reduction defines a fundamental domain, which is in fact a polyhedral cone in the space of positive-definite matrices, but the facets of this cone are known only up through seven dimensions.  (It is most naturally defined using infinitely many constraints, only finitely many of which are needed in any given dimension, but figuring out exactly which ones are needed is difficult.)
A: 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.
