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I'm looking for a reference or a strategy to recompute them that would provide me with the $10$ matrices $4 \times 4$ that are the fundamental representation of the $\mbox{sp}(4)$ group.

We know that the symplectic group Sp(4) is a real, non-compact, connected, simple Lie group with a fundamental group isomorphic to the group of integers under addition. The Lie algebra of Sp(4) is sp(4), which consists of all $4 \times 4$ matrices $A$ that satisfy $A^T J + JA = 0$ where $J$ is the standard symplectic matrix.

I'm trying to exhibit these matrices. References to an existing paper are also fine, I couldn't find one with the actual matrix values.

Rodrigo de Azevedo
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Yann TM
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    That's the split form, right? For the Lie algebra, see https://math.stackexchange.com/a/3771729/96384, type $C_n$, specialize to $n=2$. – Torsten Schoeneberg Apr 16 '23 at 15:54
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    A small note about notation: you will see $\mathfrak{sp}(n)$ used often to refer to the compact form of $\mathfrak{sp}(2n, \mathbb{C})$ (reasoning behind the $n$ being it has an n-dimensional quaternionic rep). I think it's clear you mean the split form, but good to know about these conflicting notations. Additionally you say "the" matrices but you mean a choice of basis. There are several possible ones. Perhaps some are more natural than others, but there is still a choice here – Callum Apr 16 '23 at 20:54

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