These elements of the matrices can be simplified by hand much further{roots can be cancelled and all}, yet the Fullsimplify in Mathematica doesn’t simply it completely.

The matrix is:

`{{-((p^2 + Sqrt[m^2 + p^2] p3 - p0 (Sqrt[m^2 + p^2] + p3))/ m), -(((p^2 + (m + Sqrt[m^2 + p^2]) (m - p0)) (-I p2 + Sqrt[ p^2 - p2^2 - p3^2]))/(m (m + Sqrt[m^2 + p^2]))), (1/( m (m + Sqrt[m^2 + p^2])))(-p^2 p2 + Sqrt[m^2 + p^2] p0 p2 + m (-Sqrt[m^2 + p^2] + p0) p2 - I p0 p3 Sqrt[p^2 - p2^2 - p3^2] + I p3 Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)] + I m^2 (I p2 + Sqrt[p^2 - p2^2 - p3^2]) + I m (p3 Sqrt[p^2 - p2^2 - p3^2] + Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)])), (1/( m (m + Sqrt[m^2 + p^2])))(-I (Sqrt[m^2 + p^2] - p0) p2^2 + I m^2 (p0 - p3) - I p3 (p^2 + Sqrt[m^2 + p^2] p3 - p0 (Sqrt[m^2 + p^2] + p3)) + p2 (-p0 Sqrt[p^2 - p2^2 - p3^2] + Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)]) + m (I (p0 - p3) (Sqrt[m^2 + p^2] + p3) + p2 (-I p2 + Sqrt[p^2 - p2^2 - p3^2])))}, {-(((p^2 + (m + Sqrt[ m^2 + p^2]) (m - p0)) (I p2 + Sqrt[p^2 - p2^2 - p3^2]))/( m (m + Sqrt[m^2 + p^2]))), (-p^2 + p0 (Sqrt[m^2 + p^2] - p3) + Sqrt[m^2 + p^2] p3)/m, (1/( m (m + Sqrt[m^2 + p^2])))(-I (Sqrt[m^2 + p^2] - p0) p2^2 + I m^2 (p0 + p3) + I m (Sqrt[m^2 + p^2] - p3) (p0 + p3) + I p3 (p^2 - Sqrt[m^2 + p^2] p0 - Sqrt[m^2 + p^2] p3 + p0 p3) + p0 p2 Sqrt[p^2 - p2^2 - p3^2] - p2 Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)] - m p2 (I p2 + Sqrt[p^2 - p2^2 - p3^2])), (1/( m (m + Sqrt[m^2 + p^2])))(p^2 p2 + m (Sqrt[m^2 + p^2] - p0) p2 - Sqrt[m^2 + p^2] p0 p2 + I p0 p3 Sqrt[p^2 - p2^2 - p3^2] - I p3 Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)] + m^2 (p2 + I Sqrt[p^2 - p2^2 - p3^2]) + I m (-p3 Sqrt[p^2 - p2^2 - p3^2] + Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)]))}, {(1/( m (m + Sqrt[m^2 + p^2])))(-p^2 p2 + Sqrt[m^2 + p^2] p0 p2 + m (-Sqrt[m^2 + p^2] + p0) p2 + I p0 p3 Sqrt[p^2 - p2^2 - p3^2] - I p3 Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)] + I m^2 (I p2 + Sqrt[p^2 - p2^2 - p3^2]) + I m (-p3 Sqrt[p^2 - p2^2 - p3^2] + Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)])), (1/( m (m + Sqrt[m^2 + p^2])))(I (Sqrt[m^2 + p^2] - p0) p2^2 - I m^2 (p0 + p3) + I p3 (-p^2 + Sqrt[m^2 + p^2] p0 + Sqrt[m^2 + p^2] p3 - p0 p3) + p0 p2 Sqrt[p^2 - p2^2 - p3^2] - p2 Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)] - I m ((Sqrt[m^2 + p^2] - p3) (p0 + p3) + p2 (-p2 - I Sqrt[p^2 - p2^2 - p3^2]))), (-p^2 + p0 (Sqrt[m^2 + p^2] - p3) + Sqrt[m^2 + p^2] p3)/ m, ((p^2 + (m + Sqrt[m^2 + p^2]) (m - p0)) (-I p2 + Sqrt[ p^2 - p2^2 - p3^2]))/(m (m + Sqrt[m^2 + p^2]))}, {(1/( m (m + Sqrt[m^2 + p^2])))(I (Sqrt[m^2 + p^2] - p0) p2^2 - I m^2 (p0 - p3) + I p3 (p^2 + Sqrt[m^2 + p^2] p3 - p0 (Sqrt[m^2 + p^2] + p3)) + p2 (-p0 Sqrt[p^2 - p2^2 - p3^2] + Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)]) + m (-I (p0 - p3) (Sqrt[m^2 + p^2] + p3) + p2 (I p2 + Sqrt[p^2 - p2^2 - p3^2]))), (1/( m (m + Sqrt[m^2 + p^2])))(p^2 p2 + m (Sqrt[m^2 + p^2] - p0) p2 - Sqrt[m^2 + p^2] p0 p2 - I p0 p3 Sqrt[p^2 - p2^2 - p3^2] + I p3 Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)] + m^2 (p2 + I Sqrt[p^2 - p2^2 - p3^2]) + I m (p3 Sqrt[p^2 - p2^2 - p3^2] + Sqrt[-(m^2 + p^2) (-p^2 + p2^2 + p3^2)])), ((p^2 + (m + Sqrt[ m^2 + p^2]) (m - p0)) (I p2 + Sqrt[p^2 - p2^2 - p3^2]))/( m (m + Sqrt[m^2 + p^2])), -(( p^2 + Sqrt[m^2 + p^2] p3 - p0 (Sqrt[m^2 + p^2] + p3))/m)}} `

some of the substitution we can make are (already taken),

`{p1 -> Sqrt[p^2 - p2^2 - p3^2], e -> Sqrt[p^2 + m^2]} Assuming[{Element[{p0, p, p2, p3}, Reals], m > 0}, sat2 = sat // FullSimplify] `

What more could be done to simplify the elements of the matrix?