SU(2) and SU(3) differ quite a bit.

The Lie algebra of SU(2) formed by the three generators $g_n$ is the same as the algebra formed by the SU(2) matrices/elements $F_n=e^{\pi \cdot i \cdot g_n / 2}$. In fact, the $F_n$ are just given by $F_n=i \cdot g_n$. Speaking sloppily, the $F_n$ are "finite" elements of SU(2), the $g_n$ are "infinitesimal" "in" SU(2).

Instead, for SU(3), the set $F_m$ and its algebraic relations differ from that of the eight generators $g_m$. The $F_m$ have no simple relation to the $g_m$. For example, in the usual representation, 7 of the 8 generators $g_m$ have a diagonal element that is zero; in the same representation, the $F_m$ have a 1 in that same position.

Is the set of the eight finite SU(3) elements $F_m=e^{\pi \cdot i \cdot g_m / 2}$ $-$ defined by the eight SU(3) generators $g_m$ $-$ an object of study or of teaching? Is there a simple way to think about this set? (Is it an algebra?) Is there a simple multiplication or commutation or other table for it? Is the set $F_m$, thus the eight finite elements generated by the infinitesimal generators, used in any physical or other setting? Does the set have a name?

Thank you for any help whatsoever.