In Fulton and Harris's Representation Theory: A First Course, they have the following section when describing the decomposition of $\mathfrak{sl}_3(\mathbb{C})$ representations (p. 170 in my edition):
By the description we have already given of the action of $\mathfrak{sl}_3(\mathbb{C})$ on the representation V in terms of the decomposition $V = \bigoplus_\alpha V_\alpha$, we see that the sub algebra $\mathfrak{s}_{L_1 - L_2}$ will shift eigenvalues $V_\alpha$ only in the direction of $L_2 - L_1$; in particular, the direct sum of eigenspaces in question, namely the subspace $$W = \bigoplus_k g_{\alpha + k(L_2 - L_1)}$$ of V will be preserved by the action of $\mathfrak{s}_{L_1 - L_2}$...... we may deduce from this that the eigenvalues of $H_{1,2}$ on $W$ are integral, and symmetric with respect to zero. Leaving aside the integrality for the moment, this says that the sting of dots in the diagram (12.12) must be symmetric with respect to the line $\langle H_{1,2} , L \rangle =0$
I cannot, for the life of me, seem to find the definition of $L$ earlier in the text. Can someone tell me what it is, because it seems pretty essential to what is going on?
$L$ is not a previously defined object. It is the variable for the points of the line being described. In other words, the line is the set of points $L$ such that $\langle H_{1,2}, L \rangle = 0$.