The differential of the exponential map on a symmetric space $M$ can be expanded (abusing some notation) as

$d{\rm Exp}_X=\sum_{n=0}^{\infty}\frac{({\rm ad}X)^{2n}}{(2n+1)!}.$

This is an old (1958) result of Helgason. Has anyone computed the same differential for the tangent bundle $TM$ of a symmetric space?