  We review and provide simplified proofs related to the Magnus expansion, and
improve convergence estimates. Observations and improvements concerning the
Baker--Campbell--Hausdorff expansion are also made. In this Part III, we
consider the Banach--Lie algebraic setting. We show how to improve the
"standard" convergence bound $\boldsymbol\delta= 2.1737374\ldots$ using the
customary generating function / ODE methods. Then we discuss how to achieve
better convergence bounds using the resolvent method. The emphasis here is on
the variety of the methods, rather than pushing them to the extreme.
Nevertheless, regarding the cumulative convergence radii, we show how to
establish $2.427<\mathrm C_\infty^{\mathrm{Lie}}\leq 4$ for the Magnus
expansion, and $2.93<\mathrm C^{\mathrm{Lie}}_2 \leq 4\sqrt2=5.656\ldots$ for
the BCH expansion.
