$C^{1,\alpha}$-subelliptic regularity on SU(3) and compact, semi-simple Lie groups

Let the vector fields $X_1, ... , X_{6}$ form an orthonormal basis of ${\mathcal H}$, the orthogonal complement of a Cartan subalgebra (of dimension $2$) in SU(3). We prove that weak solutions $u$ to the degenerate subelliptic $p$-Laplacian $$\Delta_{\mathcal{H},{p}} u(x)=\sum_{i=1}^{6} X_i^{*}\left(|\nabla_{\hspace{-0.1cm} {\mathcal H}} u|^{p-2}X_{i}u \right) =0,$$ have H\"older continuous horizontal derivatives $\nabla_{\hspace{-0.1cm}{\mathcal H}} u=(X_1u, \ldots, X_{6}u)$ for $p\ge 2$. We also prove that a similar result holds for all compact connected semisimple Lie groups.