Significance of Gram-Schmidt coefficients in LLL algorithm
Say you have an integer lattice $\mathcal{L}$ with (ordered) basis $B=\{{\bf b}_1,\ldots,{\bf b}_n \},$ ${\bf b}_j\in {\bf Z}^m$, $n\leq m$. As you wrote the first condition of the LLL reduced basis $\{ {\bf u}_1,\ldots,{\bf u}_n\}$ is the process of size reduction. That is $|\mu_{i,j}|\leq \frac{1}{2}$, $j&lt;i$. If you set $M_i=\mathrm{span}({\bf b}_1,\ldots,{\bf b}_i),\ i&lt;n$ then this condition shortens the projection $\mathrm{proj}_{M_i}({\bf b}_{i+1})$, while the second condition (Lovász)  shortens the projection $\mathrm{proj}_{M_i^\perp}({\bf b}_{i+1})= {\bf b}_{i+1}^*$.

References

Phong Q. Nguyen and Damien Stehlé. An LLL algorithm with quadratic complexity. Siam J. Comput., 2009. doi:10.1137/070705702 [PDF]