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I want to know a logical method of performing a commutation for [$a\hat{A} + b\hat{B}, c\hat{C} + d\hat{D}$] where $a$, $b$, $c$ and $d$ are just constants. I know the rules for [$\hat{A} + \hat{B}, \hat{C}$] but cannot seem to apply it for the former mentioned.

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Commutator brackets are linear in both slots, so $$[a\hat A+b\hat B,c\hat C +d\hat D]=ac[\hat A,\hat C]+ad[\hat A,\hat D]+bc[\hat B,\hat C]+bd[\hat B,\hat D]$$

This follows from the rule you already know. If $[\hat A+\hat B,\hat C]=[\hat A,\hat C]+[\hat B,\hat C]$, then

$$[\hat A,\hat C+\hat D]= -[\hat C + \hat D,\hat A]= -\left([\hat C,\hat A]+[\hat D,\hat A]\right) $$ $$=[\hat A,\hat C]+[\hat A,\hat D]$$

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