Some discussions seem to think the Lorentz force is incompatible with special relativity or somehow badly formulated. I thought a simple flux picture interpretation could show how it works. Note it is only for relative motion and so should transform with a Lorentz transformation.
In the image, you see how the field B presses on the field lines of the same sign on the near side of the wire and attracts the opposite flux on the far side. that gives the force as shown by the red arrow. That is indeed in the direction of L.I x B.
User posted image: http://postimage.org/image/wanhy7tln/