Exact solutions of Einstein-Yang-Mills theory with higher-derivative coupling

We construct an exact classical solution of an Einstein-Yang-Mills system in ten space-time dimensions whose action contains a term quartic in Yang-Mills field strength. The solution provides compactification proposed by Cremmer and Scherk; the ten-dimensional space-time with a cosmological constant is compactified to the direct product of four-dimensional Minkowski space-time and a six-dimensional sphere $S^6$ whose radius is determined by coupling constants. We also construct a solution which compactifies the ten-dimensional space-time to ${\mathrm{AdS}}_4\times S^6$ when the cosmological constant is absent.