Conditional finite-dimensional Lie and Strang splitting residual orders

For the optional splitting branch of the finite-dimensional coupled model, suppose $a,b\in C^3(U,{\mathbb{R}}^d)$ define the additive autonomous ODE of Additive autonomous ODE partitions, their exact subflows use the Lie and Strang compositions of Lie and Strang compositions of exact split flows, and all exact and numerical paths lie with positive margin in a common compact convex tube on $[0,T]$. Initialize both schemes at the exact initial value $y_0=y(0)$. Under the remaining Lipschitz, regularity, and step-size hypotheses of Lie and Strang splitting have global orders one and two, with separate subflow errors, the exact Lie and Strang schemes have global errors $O(h)$ and $O(h^2)$ respectively. Approximate subflows retain these orders only under the separate per-subflow error budgets and composition stability condition stated in that theorem. This corollary makes no claim for unbounded operators or general moving-domain PDEs.