In Peskin & Schroeder p.39 they introduce the 4x4-matrices
$$\left(\mathcal{J}^{\mu\nu}\right)_{\alpha\beta} = i \left(\delta^{\mu}_{\;\alpha} \delta^{\nu}_{\;\beta} - \delta^{\mu}_{\;\beta}\delta^{\nu}_{\;\alpha}\right) \tag{3.18}$$
which they "pull out of a hat" and which represent the Lorentz algebra, that is they satisfy
$$\left[ \mathcal{J}^{\mu\nu} , \mathcal{J}^{\rho\sigma} \right] = i \left( \eta^{\nu\rho} \mathcal{J}^{\mu\sigma} - \eta^{\mu\rho}\mathcal{J}^{\nu\sigma} - \eta^{\nu\sigma}\mathcal{J}^{\mu\rho} + \eta^{\mu\sigma}\mathcal{J}^{\nu\rho} \right)\tag{3.17}$$
I am having difficulties interpreting the objects $\left( \mathcal{J}^{\mu\nu} \right)_{\alpha\beta}$. In his script about QFT, David Tong describes p.82 the objects $\left(\mathcal{J}^{\mu\nu}\right)$ as the six antisymmetric matrices describing the six transformations of the Lorentz group: 3 rotations $\mathbf{J}=\left(J_x,J_y,J_z\right)$ and 3 boosts $\mathbf{K}=\left(K_x,K_y,K_z\right)$, where as I understand it each $J_i$ and $K_i$ is a matrix. Then he says that the indices $\alpha\beta$ refer to the components of the matrices $\left(\mathcal{J}^{\mu\nu}\right)$, and gives as an example
$$\left(\mathcal{J}^{01}\right)^{\alpha}_{\;\beta} = \left[ \begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right]$$
which generates boosts in the $x^1$ direction. So far so good. I am interested into inserting (3.18) in (3.17) in order to check that the matrices satisfy the Lorentz algebra, however I am confused by the four indices. It seems to me that $\left(\mathcal{J}^{\mu\nu}\right)_{\alpha\beta}$ and $\left(\mathcal{J}^{\mu\nu}\right)$ cannot be the same objects really, so how do I retrieve the latter?