Suppose we have the usual commutators ($J$=Angular Momentum, $P$=Momentum, $K$=Boosts, $H$=Hamiltonian.)
$$ 
[J_i,J_j]=i\epsilon_{ijk}J_k\quad[J_i,K_j]=i\epsilon_{ijk}K_k\quad[J_i,P_j]=i\epsilon_{ijk}P_k.
$$
and that
$$
[K_i,H]=iP_i.
$$
A professor has said that the first three relations state that $\vec J,\vec K,\vec P$ are 3-vectors and that they rotate under spatial rotations. And that the interpretation of the fourth is that if we boost energy, we get momentum.

Can anybody give the chain of logical statements, beginning with these commutators, that leads to these interpretations? This has had me confused for over a year...