Do you use a left-hand or right-hand rule with gravitomagnetism?
Usually people use a tensor formulation for the combined gravitoelectric and gravitomagnetic field, similar to the way that the Faraday tensor for electromagnetism combines the electric field and the magnetic field.
for a discussion of the Faraday tensor.
However, I suppose you could represent the gravitomagnetic field as a vector, as long as you don't try and transform it as a vector (the same caution applies to the tensor form - it "looks like" a tensor, but it doesn't transform as one).
If you did chose to represent it via a vector, I'd recommend using the right hand rule.
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