# Creating block diagonal matrix identity and one Pauli matrix

I want to create a matrix of the form:

$$X_\alpha = 1\otimes 1\otimes 1 ...\otimes X \otimes ... \otimes 1$$

where $$X$$ is the Pauli $$x$$ matrix and is the $$\alpha$$th factor, and $$1$$ is the $$2\times 2$$ identity matrix. I have found answer on how to create a block diagonal matrices of this form for a specific placement of $$X$$ but I am trying to generalize it for an arbitrary $$\alpha$$.

See this book on page 53 (section 3.4.1):

SpinQ[S_] := IntegerQ[2S] && S>=0
op[S_?SpinQ, n_Integer, k_Integer, a_?MatrixQ] /;
1<=k<=n && Dimensions[a] == {2S+1,2S+1} :=
KroneckerProduct[IdentityMatrix[(2S+1)^(k-1), SparseArray],
a,
IdentityMatrix[(2S+1)^(n-k), SparseArray]]


You'll be using S=1/2 and can simplify the above to

op[n_Integer, k_Integer, a_?MatrixQ] /;
1<=k<=n && Dimensions[a] == {2,2} :=
KroneckerProduct[IdentityMatrix[2^(k-1), SparseArray],
a,
IdentityMatrix[2^(n-k), SparseArray]]