My question is straightforward (and perhaps also dumb). I'm currently studying reductive Cartan geometries and I'm looking for a common generic matrix representation of $\mathfrak{so}(4,1)$, $\mathfrak{iso}(3,1)$ and $\mathfrak{so}(3,2)$ elements, so I can also show the splitting $\mathfrak g=\mathfrak h\oplus\mathfrak{g/h}$ in this ugly (matrix) way. The Lie algebra $\mathfrak g$ is either de Sitter, Poincare or anti de Sitter algebra, $\mathfrak h$ is the Lie algebra of the stabiliser Lie subgroup of G, H, while $\mathfrak p=\mathfrak{g/h}$ is its complement.
The reason behind this search for a common matrix rep for the above Lie algebras is that somehow I need to introduce a dimensionless parameter $\epsilon$, which takes values (1,0,-1) according to the choice of $\mathfrak g$ (de Sitter, Poincare, adS respectively). This parameter seems mandatory for defining the internal cosmological constant $\lambda$, but this is more or less a Physics issue, so I will not continue on that path here. I'm just trying to imagine how this 5x5 matrix rep should look like. I have seen such a rep here (bottom of pg. 142) but I'm not able to fully justify this expression. The author says that "in each case the Lie algebra is $\mathfrak{so}(V)$, where $V$ is a vector space with metric $(-1,+1,+1,+1,\epsilon)$". I'm assuming he is implying that $V$ is a normed vec space and this is the norm-induced metric, but still the matrix he is introducing remains a bit of a mystery to me. Could someone help me?