Equation 12.130 on Zwiebach Page 225: Is an Alpha Missing?

[ehrenfest]

Homework Statement

Shouldn't there be an

$$\alpha_{-1}\alpha_{1}$$

added to equation 12.130 on account of equation 12.129, or is that 0 for some reason?

Homework Equations

The Attempt at a Solution

 

[Jimmy Snyder]

If you work out the sum of the commutators of the terms in parentheses in equations (12.125) and (12.126), and use equation (12.128), you will find it deficient from $4L_0$ by exactly $\alpha_{-1}\alpha_1$.