The commutator form of the Sagle identity is given by one of the following
4 equivalent expressions:
[A,[B,[C,D]]] + [B,[C,[D,A]]] + [C,[D,[A,B]]] + [D,[A,[B,C]]] + [[B,D],[A,C]] = 0
[[[A,B],C],D] + [[[B,C],D],A] + [[[C,D],A],B] + [[[D,A],B],C] + [[B,D],[A,C]] = 0
[A,[[B,C],D]] + [B,[[C,D],A]] + [C,[[D,A],D]] + [D,[[A,B],C]] + [[A,C],[B,D]] = 0
[[A,[B,C]],D] + [[B,[C,D]],A] + [[C,[D,A]],B] + [[D,[A,B]],C] + [[A,C],[B,D]] = 0
All 4 versions are checked.
The first Hentzel-Peresi identity is defined as follows:
sigma{b,c,e} ([a,(c,d,e),b] + [a,b,(c,d,e)]) =
sigma{b,c,e} ([a,{{c,d},e} - {c,{d,e}},b] + [a,b,{{c,d},e} - {c,{d,e}}])
with sigma{b,c,e} the cyclic permutations over b,c,e and (c,d,e) the Jordan associator.