Liouville theorem for the Yang--Mills self-duality equations

It is shown that under certain conditions one may associatively matrix-multiply Lie-algebra-valued matrices with a componentwise Lie bracket. Using this, a simple algebraic constraint on a Lie-algebra-valued antisymmetric n x n matrix F, which in n = 4 is essentially self-duality or anti-self-duality, is described. Somewhat in analogy with Liouville's theorem for the Cauchy--Riemann equations in n = 2, it is shown that, for n>4, the constraint implies that the Lie subalgebra generated by the matrix elements /F/sub μ//sub ν/ / decomposes into copies of S-underlineO-underline(n) plus a few degenerate cases. The result may be relevant to the structure of the quantum chromodynamic vacuum