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 ?

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I actually meant the Gram matrix for a basis of the lattice, so it is both positive definite and symmetric. A change of basis matrix $U \in \textup{GL}_n(\mathbb{Z})$ acts on a Gram matrix $M$ by sending it to $UMU^t$. There are a couple of advantages of using these coordinates. One is that passing to the Gram matrix automatically mods out by the orthogonal group, and the other is that some constraints that are nonlinear in terms of a basis matrix become linear in terms of a Gram matrix. (For example, vector lengths in the lattice are linear functions of the Gram matrix entries.) –  Henry Cohn Apr 17 '11 at 12:46