We use the formula of the Lande g-factor as given in the video, with g_L = 1 and g_S = 2.
For the lowest level of the 2D multiplet we have J = 3/2, L = 2 and S = 1/2. Plugging in these values results in g_J = 4/5 = 0.8
For the highest level of the 2D multiplet L and S are the same as before, but now J = 5/2. The gives g_J = 6/5 = 1.2
I’m not sure that we can apply the same formula to the ground state of a La atom inside a lanthanum metal. In this case La is not a free atom.