# Simplifying elements of a matrix

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?