For $G = PSU(3)$, it is known that $\dim I(G;\mathbb Q) / I(G;\mathbb Q)^2 = 3$, while $H^{**}(BG;\mathbb Q)$ is obviously a power series ring in two indeterminates since $G$ has rank 2. This would would be fine except that there is an elementary argument the natural map should induce a bijection on spaces of indecomposable elements and $3\neq2$.

**What goes wrong?**

Here's the argument. Let $T$ be compact torus, $R(T)$ its representation ring, and $\widehat R(T)$ the completion with respect to the augmentation ideal $I(T)$. One has maps $$R(T) \longrightarrow \widehat R(T) \overset\sim\longrightarrow K^*(BT) \overset{\textrm{ch}}\longrightarrow H^{**}(BT;\mathbb Q),$$ where the double-star insists we view the cohomology ring as the direct product of the $H^n$. If we tensor $R(T)$ with $\mathbb Q$ beforehand, we get $$R(T;\mathbb Q) \longrightarrow \widehat R(T;\mathbb Q) \longrightarrow H^{**}(BT;\mathbb Q).$$ The last map sends a one-dimensional representation $t \in R(T)$ to $e^u$ for $u = c_1(ET \times_T \mathbb C_t)$, and hence is an isomorphism, as both rings are power series rings on $\dim T$ indeterminates and $\log t$ is a power series in $t-1$. Since $t-1 \mapsto e^u - 1 = u + u^2/2 + \cdots$, the map also sends $\hat I(T;\mathbb Q)$ to the augmentation ideal $H^{\geq 1}(BT;\mathbb Q)$ of $H^{**}(BG;\mathbb Q)$.

Now consider a compact, connected Lie group $G$ with maximal torus $T$. If we make the identification $BT = EG / T$, then $BT$ admits a right action of the Weyl group $W$ of $G$. The maps are equivariant with respect to the action of $w \in W$ since $t w$ is sent to $\exp c_1(ET \times_T \mathbb C_{t w})$ and $ET \times_T \mathbb C_{t w}$ is the pullback of $ET \times_T \mathbb C_{t}$ under $w\colon BT \to BT$. This equivariance and the standard isomorphisms $$R(T)^W = R(G),$$ $$\widehat R(T)^W = \widehat R(G),$$ $$H^{**}(BT;\mathbb Q)^W = H^{**}(BG;\mathbb Q)$$ then show $\widehat R(G;\mathbb Q) \to H^{**}(BG;\mathbb Q)$ is also an isomorphism preserving the augmentation ideal. Then, since $R(G)$ is Noetherian, the maps $$R(G;\mathbb Q) \longrightarrow \widehat R(G;\mathbb Q) \overset\sim\longrightarrow H^{**}(BG;\mathbb Q)$$ induces isomorphisms $$\frac{I(G;\mathbb Q)\phantom{2}}{I(G;\mathbb Q)^2} \overset\sim\longrightarrow \frac{\hat I(G;\mathbb Q)\phantom{2}}{\hat I(G;\mathbb Q)^2} \overset\sim\longrightarrow \frac{H^{\geq 1}(BG;\mathbb Q)\phantom{2}}{H^{\geq 1}(BG;\mathbb Q)^2}$$ of modules of indecomposables.