Whitehead product $[i_2, i_2]_W$ Let $i_2$ be the generator of $\pi_2(S^2)$ and $\eta$ be the Hopf fibration from $S^3$ to $S^2$. How would one go about showing that $[i_2,i_2]_W=2\eta$ (up to a sign)? 
This is one of the exercises in a course of topology I am currently self-studying. I have seen some solutions using the Hopf invariant, but as far as I understand, it is related to cohomology, which is the next topic. I believe that there must some way to calculate the above product from the first principles... I would like to use that to calculate the first stable homotopy group, but the only other answer I found goes the other way and uses the fact $\pi_4(S^3)=\mathbb{Z}_2$ to show $[i_2,i_2]_W=2\eta$. 

The Hopf fibration $S^1\to S^3\to S^2$ gives the long exact sequence in homotopy groups. Since higher homotopy groups of $S^1$ are trivial, we get $\pi_3(S^3)\cong\pi_3(S^2)$, with the generator given by the Hopf map. We can define the inverse isomorphism $H:\pi_3(S^2)\to \mathbb{Z}$ called the Hopf invariant as follows.
Let $x_1,x_2\in S^2$ be two distinct points and assume we pick a reprensetative of a class $\alpha\in\pi_3(S^2)$ which is a smooth map $f:S^3\to S^2$ with regular values $x_1$ and $x_2$. We can then consider $L_i:=f^{-1}(x_i)$ for $i=1,2$ which is an oriented submanifold of dimension $1$ in $S^3$, i.e. an oriented link. Define:
$$ H(\alpha):=lk(L_1,L_2)$$
the linking number of two oriented links, which is defined as the sum of pairwise linking numbers of each components of $L_1$ with each component of $L_2$. 
One then checks that this is a well-defined invariant of $\alpha$.
Since both maps are very explicit, one can calculate that:
$$H(\eta)=1$$
$$H([\iota_1,\iota_2]_W)=2$$ 
In the Hopf fibration $L_1$ and $L_2$ are two circle fibres and it is an exercise to prove that they form a Hopf link, so their linking number is $1$.
For the Whitehead product use that it is given as the composition $S^3\to S^2\vee S^2\to S^2$ of the universal Whitehead product (the attaching map of the top cell in $S^2\times S^2$) and the fold map. Both $L_1$ and $L_2$ are $2$-component links (together this forms a $0$-framed push-off of the Hopf link!) and out of four linking numbers, precisely two are $1$. To see this you might want to use that linking numbers can be computed from surfaces (here disks) that links bound in $D^4$ (see Rolfsen, for example).
