  Gravitomagnetic field is covariantly split in the {\it intrinsic} and {\it
extrinsic} parts, which are generated by rotational and translational currents
of matter respectively. The {\it intrinsic} component has been recently
discovered in the LAGEOS spacecraft experiment. We discuss the method of
detection of the {\it extrinsic} tidal component with the lunar laser ranging
(LLR) technique. Analysis of the gauge residual freedom in the relativistic
theory of three-body problem demonstrates that LLR is currently not capable to
detect the {\it extrinsic} gravitomagnetic effects which are at the ranging
level of few millimeters. Its detection requires further advances in the LLR
technique that are coming in the next 5-10 years.
