  Kapustin-Witten (KW) equations are encountered in the localization of the
topological N=4 SYM theory. Mikhaylov has constructed model solutions of KW
equations for the boundary 't~Hooft operators on a half space. Direct proof of
the solutions boils down to check a boundary condition. There are two
computational difficulties in explicitly constructing the solutions to Lie
algebra of higher rank. The first one is related to the commutation of
generators of Lie algebra. We derived an identity which effectively reduces
this computational difficulty. The second one involves the number of ways from
the highest weights to other weights in the fundamental representation. For
ADE-type gauge groups, we found an amazing formula which can be used to rewrite
the solutions of KW equations. This new formula of solutions bypass above two
computational difficulties.
