$\newcommand{\bbr}{\mathbb{R}} \newcommand{\bbc}{\mathbb{C}} \newcommand{\sof}{\mathsf{SO}(5)} \newcommand{\msof}{\mathfrak{so}(5)}$ Let $\mathsf{SO}(n):=\{A\in M_n(\bbr)\mid AA^T=I,\det A=1\}$ denote the special orthogonal group and $\mathfrak{so(n)}:=\{A\in M_n(\bbr)\mid A=-A^T\}$ denote the corresponding Lie algebra. It is well known that $\dim\mathfrak{so(n)}=\frac{n(n-1)}2$. In particular, $\msof$ can be generated by $10$ elements of $M_5(\bbr)$ and it is very easy to describe a particular set of generators.
But, I don't understand the description of the generators of $\msof$ given in this arXiv paper(section $2.10$). The author considers the following $8\times 8$ real matrices $$\gamma^i:=\begin{pmatrix}0&e^i\\-e^i&0\\\end{pmatrix},\gamma^4:=\begin{pmatrix}0&I\\I&0\\\end{pmatrix},\gamma^5:=\begin{pmatrix}I&0\\0&I\\\end{pmatrix}$$ where the $e^i$ are $4\times 4$ real matrices given as follows: $$e^1=-E_{12}-E_{34},e^2=-E_{13}+E_{24},e^3=-E_{14}-E_{23}.$$ Here, $E_{ij}$ is a $4\times 4$ anti-symmetric matrix with $1$ on the $i$th row and $j$th column. The author claims that the commutators of the above $\gamma$ matrices generate the Lie Algebra of $\sof$. I wonder how? The $\gamma$ matrices are $8\times 8$, whereas the $\msof$ is a subspace of $M_5(\bbr)$. Is there some kind of identification going on(perhaps using the isomorphism $\sof\cong \mathsf{Sp}(2)/\mathbb Z_2$)? I would just mention that the given paper is related to mathematical physics and I'm not much familiar with the Physics conventions.
Any help is appreciated. Thank you.
From a quick look at the paper, the author is treating $SO(5)$ as an abstract group, in particular it is not the same as the definition you have given (which I believe is referred to as the "defining representation" of the group). Right before equation (6) on page 2, he remarks that it is "well-known" that $SO(5)$ is isomorphic to the group of unitary $2\times 2$ matrices with quaternionic entries, hence a special type of $8\times 8$ real matrix since the quaternions are $4$-dimensional. And presumably he works with this representation of $SO(5)$ for the remainder of the paper.