Higher homotopy commutativity in localized Lie groups and gauge groups

The first aim of this paper is to study the $p$-local higher homotopy commutativity of Lie groups in the sense of Sugawara. The second aim is to apply this result to the $p$-local higher homotopy commutativity of gauge groups. Although the higher homotopy commutativity of Lie groups in the sense of Williams is already known, the higher homotopy commutativity in the sense of Sugawara is necessary for this application. The third aim is to resolve the $5$-local higher homotopy non-commutativity problem of the exceptional Lie group $\mathrm{G}_2$, which has been open for a long time.