We know that there is a cofiber sequence $S^3\xrightarrow{\eta}S^2\to\mathbb{C}\mathbb{P}^2$. It's easy to know that $\pi\_3^s(\mathbb{C}\mathbb{P}^2)=0$ so there is a surjection
$$\partial:\pi\_7^s(S^2\wedge\mathbb{C}\mathbb{P}^2)\to\pi\_7^s(\mathbb{C}\mathbb{P}^2\wedge\mathbb{C}\mathbb{P}^2)$$ by the long exact sequence of $\eta$. The former group $\pi\_7^s(S^2\wedge\mathbb{C}\mathbb{P}^2)$ is $\mathbb{Z}/12$, generated by the second hopf element. I wonder whether $\partial$ is an isomorphism?



Stable homotopy groups of complex projective plane
The question is equivalent to asking what the multiplication-by-$\eta$-map $\pi\_4\mathbb{CP}^2 \to \pi\_5 \mathbb{CP}^2$ is (which can be rewritten as $\pi\_2\mathbb{S}/\eta \to \pi\_3\mathbb{S}/\eta$). The source is generated by the lift of $2 \in \pi\_2S^2$ to $\pi\_2 \mathbb{S}/\eta$. Thus, the question translates to the question what the Toda bracket $\langle \eta, 2, \eta\rangle$ is (the translation is classical, see e.g. Lemma 4.1 in [my thesis](https://webspace.science.uu.nl/%7Emeier007/Thesis-FinalVersionb.pdf) for an account). As $\eta$ is $2$-torsion, only the $2$-local computation is relevant. As follows e.g. by a Massey product computation in the Adams spectral sequence, this Toda bracket is here the coset $2\nu + 4\pi\_3\mathbb{S}$ (see e.g. Proposition 6 in Chapter 17 of Mosher--Tangora). Thus, $2$-locally, $\pi\_4\mathbb{CP}^2 \to \pi\_5 \mathbb{CP}^2$ has image generated by $2\nu$, while $3$-locally it is zero. Thus $\partial$ is an isomorphism $3$-locally and has image $\mathbb{Z/2}$ when working $2$-locally. This implies $\pi\_3(C\eta \wedge C\eta) \cong \mathbb{Z}/6$.


In general, primary information (i.e. multiplication by elements) on a cone like $\mathbb{S}/\eta$ translates into secondary information (i.e. $3$-fold Toda brackets) on $\mathbb{S}$.


Edit: I have corrected above my earlier erroneous Toda bracket computation as caught by Nanjun Yang and corrected also the result.


