with α α ^ ^ being purely real symmetric matrices, and β β being purely imaginary skew-symmetric. In this case, purely real solutions to the equation can be found; these are the Majorana spinors. Under the action of Lorentz transformations, these transform under the (purely real) spin group Spin ⁡ ⁡ ( 1 , 3 ) . This stands in contrast to the Dirac spinors, which are only covariant under the action of the complexified spin group Spin C ⁡ ⁡ ( 1 , 3 ) . The interpretation is that complexified spin group encodes the electromagnetic potential, the real spin group does not.