I've looking for the matrices of the O(3, 2) Lie group or, instead of them, the structure constants of its algebra $\mathfrak o(3, 2)$ but I've not been successfull. Does anyone know them or can suggest me a book with these contents?
Thanks in advance! ;)
Maybe what is meant here is the (real) orthogonal group of matrices which leave a symmetric bilinear form of signature $(3,2)$ invariant. Compare https://en.wikipedia.org/wiki/Indefinite_orthogonal_group. This gives a description in terms of matrices as (in this case)
$O(3,2) = \lbrace A \in M_5(\Bbb R) \text{ such that } gA^{tr}g =A^{-1} \text{ where } g = diag[1,1,1,-1,-1] \rbrace$
About good book references, I have found S. Helgasson, Differential geometry, Lie groups, and symmetric spaces as well as Onishchik/Vinberg, Lie Groups and Algebraic Groups very helpful.
By the way, since a signature $(3,2)$ implies a Witt index of $2$, which is maximal in a five-dimensional space, I'm quite sure that this is actually the "split" orthogonal group of dimension 5, with root system $B_2$. So the structure constants would be the same (if given as integers, e.g. for a Chevalley basis) as those for the complex group $O(5, \Bbb C)$.
Added 17-X-2018:
I can offer two matrix representations of the split Lie algebra $\mathfrak{so}_{3,2}(k)$ over an arbitrary characteristic zero field $k$. However, both use a different normalisation than above: Namely, if to represent the bilinear form, we do not use the above diagonal matrix, but the matrix (EDIT: corrected)
$G := \pmatrix{0&0&1&0&0\\ 0&0&0&1&0\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&0&0&1}$
(which is equivalent, but amounts to a different choice of basis), then the (10-dimensional) Lie algebra is given by
$\lbrace \pmatrix{a&b&0&e&g\\ c&d&-e&0&h\\ 0&f&-a&-c&i\\ -f&0&-b&-d&j\\ -i&-j&-g&-h&0\\} : a, ..., j \in k \rbrace$
and you can pick a letter, set that one $=1$, the others $=0$, and then compute the relations between them; there you have structure constants for that basis. (I found that in my thesis, p.85.)
On the other hand, Bourbaki gives yet another normalisation for that split Lie algebra in exercise 4 to paragraph 13 of chapter VIII of the Lie Groups and Algebras book. There, they instead write the bilinear form with the matrix
$H := \pmatrix{0&0&0&0&1\\ 0&0&0&1&0\\ 0&0&1&0&0\\ 0&1&0&0&0\\ 1&0&0&0&0}.$
Using this, it describes the Lie algebra as those $5\times 5$-matrices which are skew-symmetric w.r.t. the second diagonal, i.e.
$\lbrace \pmatrix{a&b&c&d&0\\ e&f&g&0&-d\\ h&i&0&-g&-c\\ j&0&-i&-f&-b\\ 0&-j&-h&-e&-a\\} : a, ..., j \in k \rbrace$
Again, one can compute the structure constants for the obvious ten basis elements here. Actually, if you do that for either of these two matrix sets, I would be interested in seeing them.