Let $G$ be a semisimple split algebraic group, $T$ its split maximal torus and $W$ corresponding Weyl group. Let $T^*$ denote the character lattice of $T$ and $\Lambda$ denote the weight lattice, so $T^*\subseteq \Lambda$.
Consider integral group rings $\mathbb{Z}[T^*]\subseteq\mathbb{Z}[\Lambda]$. So $\mathbb{Z}[T^*]$ is additively generated by $e^{\chi},\chi\in T^*$ and $\mathbb{Z}[\Lambda]$ is additively generated by $e^{\lambda},\lambda\in\Lambda$. Consider the augmentation map $aug\colon \mathbb{Z}[\Lambda]\to\mathbb{Z}$ defined by $e^{\lambda}\mapsto 1$. Let $\tilde I$ denote the kernel of $aug$ and $I=\mathbb{Z}[T^*]\cap\tilde I$. Let $\tilde I^W$ denote the ideal in $\mathbb{Z}[\Lambda]$ generated by $W$-invariant elements of $\tilde I$ and $I^W$ denote the ideal in $\mathbb{Z}[T^*]$ generated by $W$-invariant elements of $I$.
The question is: in what cases the following ideals coincide in $\mathbb{Z}[T^*]$? $$I^W+I^3=(\tilde I^W\cap\mathbb{Z}[T^*])+I^3$$ where $I^3$ is just the multiple $I\cdot I\cdot I$as ideals inside
