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Let $(M,F)$ be a connected Finsler space. An isometry of $(M,F)$ is called aClifford-Wolf translation (or simply CW-translation) if it moves all points thesame distance. The compact Finsler space $(M,F)$ is called restrictivelyClifford-Wolf homogeneous (restrictively CW-homogeneous) if for any twosufficiently close points $x_1,x_2\in M$, there exists a CW-translation$\sigma$ such that $\sigma(x_1)=x_2$. In this paper, we define the goodnormalized datum for a homogeneous non-Riemannian $(\alpha,\beta)$-space, anduse it to study the restrictive CW-homogeneity of left invariant$(\alpha,\beta)$-metrics on a compact connected semisimple Lie group. We provethat a left invariant restrictively CW-homogeneous $(\alpha,\beta)$-metric on acompact semisimple Lie group must be of the Randers type. This gives a completeclassification of left invariant $(\alpha,\beta)$-metrics on compactsemi-simple Lie groups which are restrictively Clifford-Wolf homogeneous.

Clifford-Wolf homogeneous left invariant $(\alpha,\beta)$-metrics on compact semi-simple Lie groups