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

The gravitomagnetic influence on Earth-orbiting spacecrafts and on the lunar orbit