@frappat]. In the Cartan-Weyl basis, the former can be written as $K_0$, $K_{\pm}$ and $Y$, while the latter are denoted by $V_{\pm}$ and $W_{\pm}$. They satisfy the following (nonvanishing) commutation or anticommutation relations $$\begin{array}{ll}
  [K_0, K_{\pm}] = \pm K_{\pm}, &\qquad  [K_+, K_-] = - 2K_0, \\[0.3cm]
  [K_0, V_{\pm}] = \pm \tfrac{1}{2} V_{\pm}, &\qquad [K_0, W_{\pm}] = \pm \tfrac{1}{2} W_{\pm}, 
  \\[0.3cm]
  [K_{\pm}, V_{\mp}] = \mp V_{\pm}, &\qquad [K_{\pm}, W_{\mp}] = \mp W_{\pm}, \\[0.3cm]
  [Y, V_{\pm}] = \tfrac{1}{2} V_{\pm}, &\qquad [Y, W_{\pm}] = - \tfrac{1}{2} W_{\pm}, \\[0.3cm] 
  \{V_{\pm}, W_{\pm}\} = K_{\pm}, &\qquad \{V_{\pm}, W_{\mp}\} = K_0 \mp Y,      
\end{array}  \label{eq:def-1}$$ together with the Hermiticity properties $$K_0^{\dagger} = K_0, \qquad K_{\pm}^{\dagger} = K_{\mp}, \qquad Y^{\dagger} = Y, \qquad 
  V_{\pm}^{\dagger} = W_{\mp}.  \label{eq:def-2}$$ From (\[eq:def-1\]), it is clear that $K_0$ and $Y$ are the weight generators, while $K_-$, $V_-$, $W_-$ (resp. $K_+$, $V_+$, $W_+$) are the lowering (resp. raising) generators.

A well-known realization of this superalgebra uses $\nu$ commuting pairs of bosonic and fermionic creation and annihilation operators, ${a^{\dagger}}_i$, $a_i$ and ${b^{\dagger}}_i$, $b_i$, $i=1$, 2, …, $\nu$, where $[a_i, {a^{\dagger}}_j] = \delta_{ij}$ and $\{b_i, {b^{\dagger}}_j\} = \delta_{ij}$, respectively. Such a realization is given by $$\begin{split}
  & K_0 = \frac{1}{2} \biggl(\sum_i {a^{\dagger}}_i a_i + \frac{\nu}{2}\biggr), 