Does there exist a lattice in $SL(n,\mathbb{R})$ which does not contain any $\mathbb{R}$-diagonalizable copy of $\mathbb{Z}^{n-1}$ ?
I know that the answer is no if the lattice is supposed to be uniform, and that $SL(n,\mathbb{Z})$ also satisfies this property, that is why I wonder if every lattice satisfies this property.