Splitting of uniform bundles on quadrics

We show that there exist only constant morphisms from Q²ⁿ⁺¹(n≥ 1) to G(l,2n+1) if l is even $(0<l<2n)$ and $(l,2n+1)$ is not $ (2,5)$. As an application, we prove on Q²ᵐ⁺¹ and Q²ᵐ⁺²(m≥ 3), any uniform bundle of rank $2m$ splits.