A fourteen-dimensional biperfect Lie bialgebra

We construct a fourteen-dimensional Lie bialgebra for which both the underlying Lie algebra and its dual are perfect. The construction is defined over $\mathbb{Q}$ and is self-dual over $\mathbb{C}$. It gives an affirmative answer to Question 4.1(3) of Etingof–Gelaki–Guralnick–Saxl, arXiv:9912068. Its twenty-eight-dimensional Drinfeld double is quasitriangular with injective cobracket, also answering Question 4.1(4).