Let $\Phi$ be the corresponding root system of a finite dimensional semisimple Lie algebra. Let $\alpha_i,\alpha_j\in \Phi$ be simple roots. I want to find all Lie algebras which satisfy
$$\langle \alpha_i^\vee,\alpha_j\rangle\langle \alpha_i,\alpha_j^\vee\rangle\alpha_i-3\alpha_i=\delta_{ij}\alpha_j$$
So if $i=j$, then
$$\langle \alpha_i^\vee,\alpha_i\rangle\langle \alpha_i,\alpha_i^\vee\rangle\alpha_i-3\alpha_i=\alpha_i\Rightarrow \langle \alpha_i^\vee,\alpha_i\rangle^2=4$$
Which doesn't tell us much, as this is satisfied for any Lie algebra. However if $i\neq j$, then
$$\langle \alpha_i^\vee,\alpha_j\rangle\langle \alpha_i,\alpha_j^\vee\rangle\alpha_i-3\alpha_i=0\Rightarrow \langle \alpha_i^\vee,\alpha_j\rangle\langle \alpha_i,\alpha_j^\vee\rangle=3$$
Which is (as far as I can tell) $G_2$.
Now what bothers me, is that I'm suppoed to find all Lie algebras which satisfy this. So to me it seems that there should something else than $G_2$. Are there any other?
Best Answer
Even with the comment this is still not entirely clear. The only way I can make sense of the question is as follows:
Let $\Phi$ be a root system and $\lbrace \alpha_1, ..., \alpha_n \rbrace=\Delta \subset \Phi$ (note $n = rank(\Phi)$) a chosen set of simple roots. Let $$\langle \alpha_i^\vee,\alpha_j\rangle\langle \alpha_i,\alpha_j^\vee\rangle\alpha_i-3\alpha_i=\delta_{ij}\alpha_j$$ for all $\alpha_i, \alpha_j \in \Delta$. Then what can we say about $\Phi$?
The answer is that $\Phi$ must be the root system of type $A_1$ (with $n=1$) or of type $G_2$ (with $n=2$), and no other. Namely, for $n=1$ there is only that root system (and with your reasoning it satisfies the criterion trivially); as soon as $n\ge 2$, with your reasoning we see that e.g. the angle between the roots $\alpha_1, \alpha_2$ is $5\pi/6$ or $\pi/6$; it must be the first option because otherwise they could not both be simple roots; so they span a root system of type $G_2$. Now every root system is a direct sum of irreducible ones; and by inspection, none of the other irreducible ones contains a subsystem of type $G_2$. This means that if the rank of $\Phi$ is $\ge 3$, it must be of the form $G_2 \oplus \Phi'$ with some other root system $\Phi'$ of rank $n-2$. However, then any of the roots $\alpha_3, ..., \alpha_n$ are in $\Phi'$, hence orthogonal to both $\alpha_1, \alpha_2$, in particular the above criterion is not satisfied e.g. for $i=1, j=3$ where $\langle \alpha_i^\vee,\alpha_j\rangle\langle \alpha_i,\alpha_j^\vee\rangle =0$, contradiction.