Solving the matrix exponential function for the Lie groups SU(3), SU(4) and Sp(2)

The well known analytical formula for $SU(2)$ matrices $U = \exp(i \vec \tau \!\cdot\! \varphi\,) \cos|\vec \varphi\,| + i\vec \hat\varphi \, \sin|\vec \varphi\,|$\\ is extended to the $SU(3)$ group with eight real parameters. resulting involves sum over three roots of a cubic equation, corresponding so-called irreducible case, where one has employ trisection an angle. When going special unitary $SU(4)$ 15 prameters, four quartic equation. associated resolvent equation positive belongs again case. Furthermore, by imposing pertinent condition on can also treat symplectic $Sp(2)$ ten Since there occur as two pairs opposite sign, this simplifies considerably. An outlook situation formulas $SU(5)$, $SU(6)$ and $Sp(3)$ given.