  We show a detailed investigation of the split Kondo effect in a carbon
nanotube quantum dot with multiple gate electrodes. It is found that the
splitting decreases for increasing magnetic field, to result in a recovered
zero-bias Kondo resonance at finite magnetic field. Surprisingly, in the same
charge state, but under different gate-configurations, the splitting does not
disappear for any value of the magnetic field, but we observe an avoided
crossing of two high-conductance lines. We think that our observations can be
understood in terms of a two-impurity Kondo effect with two spins coupled
antiferromagnetically. The exchange coupling between the two spins can be
influenced by a local gate, and the non-recovery of the Kondo resonance for
certain gate configurations is explained by the existence of a small
antisymmetric contribution to the exchange interaction between the two spins.
