You can split the 15 generators of SU(4) into 5 groups of SU(2) generators, below are two ways to do that. The table used below is a product table, the generators in the bottom right square are the products of the first row with the first column.Each of the 5 groups is an anti-commuting triplet. Each of them can be used as a base of SU(2).
This uses the 6 generators of SO(4) below, SO(4) has two anti-commuting triples while the triples commute between each other.
The SO(4) matrices are given below. Note that the colored 3x3 sub matrices are the SO(3) rotation matrices.
Both SO(4) triples are also alternative bases for quaternians.
Where $*$ is complex conjugation. Further, to get the x,y,z correspondence to the $SO(3)$ generators as mentioned above we have re-associated the x,y,z coordinates with the Pauli matrices as follows: $\sigma^x\!=\!\sigma^3,~\sigma^y\!=\!-\sigma^2,~\sigma^z\!=\!\sigma^1$
Note that this representation is much cleaner as some horrible ones that extend SU(3) to SU(4)
The full table written out gives: