The following part is taken from the Algebraic Bethe Ansatz paper https://arxiv.org/abs/hep-th/9605187 (Page 7).
The definition of the Lax operator involves the local quantum space $h_n$ and the auxiliary space $V$, which for the beginning will be also $\Bbb{C}^2$. Lax operator $L_{n,a}(\lambda)$ acts in $h_n \otimes V$ and is given explicitly by the expression $L_{n,a}(\lambda) = \lambda \mathrm{I_n} \otimes \mathrm{I_a}+ i \sum_\alpha S^\alpha_n \otimes\sigma^\alpha$, where $\mathrm{I_n}$, $S^\alpha_n$ acts in $h_n$ and $ \mathrm{I_a}, \sigma^\alpha$ are unit and Paulimatrices in $V=\Bbb{C}^2; \lambda $ is a complex parameter, usually called the spectral parameter, reminding its role as an eigenvalue in the original Lax operator. Alternatively $L_{na}(\lambda)$ can be written as a 2 $\times 2$ matrix $L_{na}(\lambda)= \begin{bmatrix} \lambda + i S_n^3&iS_n^-\\iS_n^+ &\lambda-iS_n^3 \end{bmatrix}$, acting in $V$ with entries being operators in quantum space $h_n$.
After giving the explicit form of Lax operator as tensor product in the respective spaces, the alternative form is given in matrix form with the matrix entries being operators in the the first space. How exactly do we arrive in this form (The matrix which acts only on one space)? I tried checking in lots of other related paper but clearly I'm missing something here. Can anybody please give me a hint or a direction?
Both forms of the $L_{na}(\lambda)$ are acting on the same space $h_n \otimes V$. The first makes it obvious by writing it as a sum of tensor products of operators acting on each space. The second is using the standard basis to write it as a block matrix.
Take the top left entry. That corresponds to the part of the matrix that is takes vectors of the form $|x0\rangle \in h_n \otimes V$ to $|y0\rangle$. So a 2 by 2 matrix for that entry whose $xy$ entry is described that matrix element. $\langle x0\mid L_{na}(\lambda) | y0\rangle = \lambda \langle x |I_n| y \rangle \langle 0 | I_V | 0 \rangle + i \sum_\alpha \langle x | S_n^\alpha | y \rangle \langle 0 | \sigma^\alpha | 0 \rangle$. You can see that this gives the entries for the operator $\lambda I_n + i S_n^3$ acting on $h_n$.
Do the same for the other three entries by switching around some of the $|0\rangle$'s above to $|1\rangle$'s.