Chen identity ($S_N(x{)}_{s,u}=S_N(x{)}_{s,t}\otimes S_N(x{)}_{t,u}$)

Partition each ordered integration simplex according to the number of times before the concatenation point. The integral over each resulting region factors into an initial iterated integral and a final iterated integral. Summing these factorizations proves the identity in the truncated tensor algebra.