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Given supersmmetry transformation:

$$δX^\mu = i\barξψ^\mu.δψ^\mu = ξγ^a(∂_aX^\mu-\frac i2\barχ_a ψ^\mu). δχ_a = 2 D_aξ.$$

How to calculate the transformation algebra of $X^\mu$ and $\psi^\mu$?

Does that mean calculating $\{δ_1,δ_2\}X^\mu$ and $\{δ_1,δ_2\}\psi^\mu$?

Reference: Superstring theory (by GSW), Section $4.1.1$ Global World-Sheet Symmetry.

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It just means calculating $[δ_1,δ_2]X^\mu$ and $[δ_1,δ_2]\psi^\mu$... Perhaps you should first practice with susy invariance on a simple scalar multiplet, no doubt covered in your relevant course, and only then apply your expertise to this one.

Recall that $\delta \phi \equiv i[\bar \xi Q,\phi]$ for an arbitrary field $\phi$, boson or fermion. So δ may be thought of as a boson operator, and it is its commutators with other fields or operators that are meaningful, not its anticommutators. The fermionic nature of the supercharge Q is neutralized by the fermionic parameter ξ to yield a bosonic δ. Confirm this is so by comparing the l.h.s. of your transformations to the r.h.sides, and their respective statistics.

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