LULU is syntomic

Let $G$ be a Chevalley group over a field $\Bbbk$. Fix a maximal torus $\mathbb{T}$ in $G$, along with opposite Borel subgroups $B$ and $B^-$ satisfying $\mathbb{T} = B \cap B^-$, and denote by $\mathbb{U} := R_u(B)$ and $\mathbb{L} := R_u(B^-)$ their respective unipotent radicals. We prove that the multiplication map $\mu \colon \mathbb{L} \times \mathbb{U}\times \mathbb{L} \times \mathbb{U} \longrightarrow G$ is syntomic and faithfully flat over any base field $\Bbbk$.