Suppose you've been given the simple roots of a Lie algebra. When finding the remaining roots, do you need to check the root string of all the roots through all the other roots, just simple roots through the simple roots, or just the simple roots through all other roots?
I ask as I've been given the following question: "The rank two Lie algebra of the 14-dimensional simple group $G_2$ has simple roots $\alpha_1 = (1,0)$ and $\alpha_2=(-3,\sqrt3 )/2 $ (in a Cartesian basis). Use the following information to sketch the root system for $G_2$."
I find that string length of $\alpha_1$ through $\alpha_2$ to be 4, hence $\alpha_2, \alpha_1 + \alpha_2, 2\alpha_1+\alpha_2$ and $ 3\alpha_1+\alpha_2$ are roots. Then the string length of $\alpha_2$ through $\alpha_1$ to be 2 which yields no new roots. I then check $\alpha_1$ through $ 3\alpha_1+\alpha_2$, which gives 2. Hence one new root. But $ 3\alpha_1+\alpha_2$ through $\alpha_1$ is length 2 and hence gives a new root too. This gives me seven roots in total, and Wikipedia is telling me there are only 6 positive roots in total. What have I done wrong?
Any help is appreciated!
Draw a picture!
You started out ok - the $\alpha_1$-string via $\alpha_2$ gives the listed 4 positive roots. But then it is automatically pointless to try $\alpha_1$-strings thru any of those roots - they will automatically be substrings of this string. For example your $\alpha_1$-string thru $3\alpha_1+\alpha_2$ gives just the same string in reverse order. The $\alpha_2$-string thru $3\alpha_1+\alpha_2$ does give you one more positive root, namely $3\alpha_1+2\alpha_2$. May be it was a typo and you meant $\alpha_2$?
The last step I don't understand at all. The $(3\alpha_1+\alpha_2)$-string thru $\alpha_1$ does not give any new positive roots. The roots in that string are $\alpha_1$ and $-2\alpha_1-\alpha_2$, and we dismiss the negative roots here.
You get all the roots by considering $\alpha$-strings through the previous generated positive roots, where $\alpha$ runs over the set of simple roots. You do need to apply this recursively!