Trying to find the double dot product of a Rank 4 and Rank 2 tensor

Question

I understand this question has been asked before, or something very similar to what I'm trying to accomplish at the very least, however I didn't understand much of what was happening in the other solutions. I'm trying to find the double contraction of A, a rank 4 tensor, and B, a rank 2 tensor (a matrix), or A:B, and I'm not sure how to go about doing this on Mathematica. I know that the double contraction (or double dot product) is meant to yield a rank 2 tensor. The equation that I was given and need to solve for does not indicate any indices so I'm not sure how to go about it.

    A = KroneckerDelta[9,9]
    B = 0.5*IdentityMatrix[3]

I have tried:

    Dot[A,B] (*which makes no sense*)
    A*B (*makes absolutely no sense, but I am desperate at this point*)
    Tr[A,Transpose[B]] (*But I think this only works for rank 2 tensors*)

Edit: Here is a bit more context. I'm trying to find the double dot product of the projection tensor P and a matrix which are denoted by the following:

    I = Array[KroneckerDelta, {3,3}]; 
    J = Array[KroneckerDelta,{9,9}];
    A = J -((1/3)*TensorProduct[I,I]);
and B = 0.5*IdentityMatrix[3];

Yeah. Just need help going about this, any would be great. Have a great day

Answer 1

What $A:B$ is supposed to mean is hard to tell without context (I consider it bad notation anyways because I was told in German primary school that "$:$" means division). But my guess is the following:

n = 20;
A = RandomReal[{-1, 1}, {n, n, n, n}];
B = RandomReal[{-1, 1}, {n, n}];

a = TensorContract[TensorProduct[A, B], {{3, 5}, {4, 6}}]; // AbsoluteTiming // First 

1.63814

More performant variants are however these:

b = Activate@TensorContract[Inactive[TensorProduct][A, B], {{3, 5}, {4, 6}}]; // AbsoluteTiming // First 
c = ArrayReshape[A, {n, n, n^2}].Flatten[B]; // AbsoluteTiming // First 

0.06622

0.000292

All of them lead to eseentially the same result:

Max[Abs[a - b]]
Max[Abs[b - c]]

8.88178*10^-15

0.