For all rank two tensors, is $A:BC = AB^T:C$?

Question

Suppose that $A$, $B$, $C$ are rank two tensors that are not necessarily symmetric, and I have a contraction as below. Is the following equivalent?

$A:BC \equiv AB^T:C$

If not, what is the correct expression that will have only C on the right side? Thanks

Answer 1

The Frobenius (aka double-dot) product of two matrices can be written in terms of the trace $$A:B = {\rm tr}(A^TB)$$ The properties of the trace give rise to various rules for rearranging terms within the product.
For example $$\eqalign{ A:BC &= AC^T:B \cr &= B^TA:C \cr &= BC:A \cr &= A^T:(BC)^T \cr }$$