Let $G$ be a reductive group over a field $k$. Let $P$ be a parabolic $k$-subgroup of $G$ and $P = L \ltimes U$ be a Levi decomposition. Is it true that every maximal split torus of $P$ lies in $L$ ? On the one hand, we have the projection map $P \mapsto P/U$ which identifies maximal split tori in $P$ with maximal split tori in $P/U$ (because $U$ is unipotent). Moreover, the same map restricted to $L$ is an isomorphism of $k$-groups $L \overset{\sim}{\to} P/U$, and therefore identifies maximal split tori on both sides. Doesn't this imply that $L$ and $P$ have the same maximal split tori ?
This confuses me because it seems to me like there should be more in $P$ than in any given Levi factor $L$. For example, if $G = GL_n$ and $P$ is the stabiliser of some subspace $V \subset k^n$, then maximal split tori in $P$ correspond to decompositions of $k^n$ into lines $D_1, \dots, D_n$ such that a subset of these lines span $V$. On the other hand, Levi factors in $P$ correspond to splittings $k^n = V \oplus W$. Therefore, a maximal split torus in a Levi factor corresponding to a decomposition $k^n = V \oplus W$ should correspond to a decomposition of $k^n$ into lines $D_1, \dots, D_n$ such that a subset of these lines span $V$ and the complement spans $W$ (or am I mistaken here ?)
I must be missing something because these two statements seem to contradict each other (I suspect that my example is incorrect, but I don't understand how).