It is well known that the fundamental class of a compact Lie group $G$ is stably spherical (see "H-Spaces and Duality" by Browder and Spanier, or "Thom Complexes" by Atiyah), and there is a stable equivalence $G\simeq A\vee S^n$ for some subcomplex $A$, with $n$ being the dimension of $G$.

My question relates to exactly when the attaching map of the top cell becomes trivial? How many suspensions are required for the top cell to split off? Is there an exact answer, or bounds for some special cases perhaps?

As an example, Mimura shows in "On the Number of Multiplications on $SU(3)$ and $Sp(2)$" that this occurs for $Sp(2)$ after exactly two suspensions, $S^2\wedge Sp(2)\simeq S^5\cup e_9\vee S^{12}$, and for $SU(3)$ after exactly three suspensions, $S^3\wedge SU(3)\simeq S^6\cup e_8\vee S^{11}$. These are very special cases. Are more general results or estimates available?