Let $G$ be a connected reductive group of an arbitrary field $k$, $P$ a parabolic $k$-subgroup, and $U = \mathscr{R}_u(P)$ the unipotent radical of $P$ (so $U$ is what is unfortunately traditionally denoted as $N$). 

For any Levi $k$-subgroup $M$ of $P$ there is a 1-parameter $k$-subgroup $\lambda:{\rm{GL}}_1 \rightarrow G$ such that $P = P_G(\lambda)$, $M = Z_G(\lambda)$, and
$U = U_G(\lambda)$.  In particular, $\lambda$ factors through the maximal split central $k$-torus $S_M$ in $M$, so $\lambda(t) \in S_M(k)$ for any $t \in k^{\times}$.  By the Borel-Tits structure theory, if $S$ is a maximal split $k$-torus in $M$ containing $S_M$ then it is also maximal as such in $G$ and the set $\Phi(G,S)$ of nontrivial $S$-weights on ${\rm{Lie}}(G)$ is a root system (spanning ${\rm{X}}(S)_{\mathbf{Q}}$), possibly non-reduced.

The $k$-group $U$ is directly spanned in any order by the root groups $U_a$ for non-multipliable $a \in \Phi(G,S)$ such that $\langle a, \lambda \rangle > 0$.  (Of course, $\dim U_a$ might be very large, and $U_a$ might be non-commutative if $a$ is divisible.)

If $a$ is not divisible in $\Phi(G,S)$ then $U_a$ is a vector group admitting a linear structure relative to which the effect of $\lambda(t)$-conjugation on $U_a$ is scaling by $t^{\langle a, \lambda\rangle}$, and is an extension of such by another using $2a$ when $a$ is divisible. In this way we see that any $t \in k^{\times}$ with $|t| < 1$ yields $s := \lambda(t) \in S_M(k)$ that does the job (as $\langle a, \lambda \rangle > 0$ precisely for those $a$ occurring as $S$-weights on ${\rm{Lie}}(U)$).