 Introduction to Smooth Manifolds, second edition by John M. Lee

Second edition, Graduate Texts in Mathematics 218, Springer. DOI: doi.org/10.1007/978-1-4419-9982-5

Inspected academic course copy: julianchaidez.net/materials/reu/lee_smooth_manifolds.pdf

Theorem 9.44 and the paragraph immediately preceding it define local flow commutation on entire time rectangles. The text explicitly distinguishes this from merely requiring both endpoint compositions to exist for incomplete vector fields.