Consensus control in multi-agent systems is a critical problem in applications such as satellite formations and aerial vehicle swarms. Attitude consensus control is the control strategy that aligns the orientations of all rigid bodies which share information according to a network, despite challenges such as communication limitations and differences in initial conditions. The control of each agent in a network requires embedded software that contains attitude control algorithms. These controllers can be designed to use rotation matrices as the attitude coordinate in the Lie group SO(3), a three-dimensional compact manifold. In this paper, existing techniques for attitude control on SO(3) are reviewed for a single rigid body and for the problem of multi-agent attitude synchronization on SO(3)^N, and subsequently are extended to the problems of configuration control on the Lie group SO(4), a four-dimensional compact manifold, and multi-agent consensus control on SO(4)^N. For configuration control of a single agent, a control law is proposed with almost global asymptotic stability on SO(4). For communication topologies such as a ring graph which results in only local stability of the consensus manifold when using the base case control law on SO(4)^N,  multi-agent configuration synchronization with feedback reshaping is used to destabilize nonconsensus equilibria and produce consensus with almost global stability on SO(4)^N. Simulations are conducted to illustrate these advantages of the proposed control strategies. Results suggest the feedback reshaping strategy previously used on SO(3)^N is justifiably extended to the problem of consensus on SO(4)^N.