We define \begin{equation} \sum_{i=1}^{4^l} P_i \otimes P_i, \sum_{i=1}^{4^m} Q_i \otimes Q_i, \end{equation} where $P_l$ is the $n$ qubit Pauli string and $Q_m$ is the $m$ qubit Pauli string.
Does the following equality hold? \begin{equation} \sum_{i=1}^{4^l} P_i \otimes P_i \otimes \sum_{i=1}^{4^m} Q_i \otimes Q_i = \sum_{i=1}^{4^{l+m}} O_i \otimes O_i, \end{equation} where $O_i$ is the $l+m$ qubit Pauli string.
Would the summation be the same although the elements of the summation are different? In fact, they should be the same.