Explicit 4-term commutator $[A,B,C,D]$ The commutator of two terms is: $$ [A,B]=AB-BA $$ The commutator of three terms is: $$ [A,B,C]=ABC+BCA+CAB-BAC-CBA-ACB $$ I was not able to google the four term commutator, not am I able to construct it from the general definition. For reference, can any one explicitly state the expression of the four term commutator?

Based on the formulas you've presented, I suspect that the general formula is $$ [A_1,A_2,\dots,A_n] = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma)\prod_{i=1}^n A_{\sigma_i}, $$ which is a formula analogous to the Leibniz formula for the determinant. Thus, your four-fold commutator would be a sum of 24 terms, 12 with a $+$ and 12 with a $-$.

The terms with a $+$ will have indices in the order $$ 1234,3124,2314,4132,2431,4213,3241,1423,1342,2143,3412,4321 $$ (corresponding to all elements of the alternating group $A_4$) and the remaining orders of indices, namely $$ 2134,1324,3214,1432,4231,2413,2341,4123,3142,1243,4312,3421 $$ will have a $-$.