I am interested in finding a general rule (from the matrix point of view) for calculating the partial trace. Starting from a matrix $$ A = X_1 \otimes X_2 \otimes \cdots \otimes X_n $$ I know how to take the partial trace for the case of only two $2\times 2$-matrices $X_i$ : $$ \def\Tr{\mathrm{Tr}}\Tr(A)_1 = \begin{pmatrix} A_{1,1} + A_{2,2} & A_{1,3} + A_{2,4}\\ A_{3,1} + A_{4,2} & A_{3,3} + A_{4,4} \end{pmatrix} $$ or element $(i,j)$ of the partial trace is the sum of elements $(i,j)$ of the diagonal blocks.
and with respect to the other subsystem $$ \bigl(\Tr(A)_1\bigr)_{i,j} := \Tr({\text{$(i,j)$-block of $A$}}) $$ Question: How can I generalize this in a simple way for $n$ matrices $X_i$ ?
Notice that if $ A = \sum_{l=1}^m X_1^l \otimes X_2^l \otimes \cdots \otimes X_n^l $ then $tr_i(A)=\sum_{l=1}^m tr(X_i^l)X_1^l \otimes X_2^l \otimes \cdots X_{i-1}^l \otimes X_{i+1}^l \otimes \cdots \otimes X_n^l $.
Now, $tr(X_i^l)=\sum_{s=1}^ne_s^tX_i^le_s$, where $\{e_1,\ldots,e_n\}$ is the canonical basis of $\mathbb{C}^n$, where $\mathbb{C}^n$ is the set of column vectors with $n$ complex entries and $X_i^l$ is a complex matrix of order $n$.
Thus, $tr(X_i^l)X_1^l \otimes X_2^l \otimes \cdots X_{i-1}^l \otimes X_{i+1}^l \otimes \cdots \otimes X_n^l=\sum_{s=1}^n L_s^t(X_1^l \otimes\cdots \otimes X_n^l)L_s$, where $L_s= Id_1 \otimes Id_2 \otimes \cdots Id_{i-1}\otimes e_s \otimes Id_{i+1} \otimes \cdots \otimes Id_n$ and $Id_t$ is the identity on the system $t$.
Finally, $tr_i(A)=\sum_{s=1}^nL_s^tAL_s$.