# Three dimensional visualization of a qutrit

My question is in reference to the paper "Three dimensional visualization of a qutrit"(https://arxiv.org/abs/1601.07361). The author's start with a symmetric two qubit density matrix written in the form $$\rho = \frac{1}{4}\left(I\otimes I +\sum_{j}(a_j\sigma_j\otimes I + a_jI\otimes \sigma_j)+ \sum_{j,k}T_{jk}\sigma_j\otimes\sigma_k\right)$$, $$(j,k=x,y,z)$$ where the $$I$$ is the $$2X2$$ identity matrix and $$\sigma_j$$'s are the familiar pauli matrices. The two qubit density matrix is therefore a $$4X4$$ density matrix. After which the author's go on define some two qubit operators as follows: And then they choose an alternate representation of $$3X3$$ spin-1 operators in order to express the $$4X4$$ two qubit density matrix as a $$3X3$$ density matrix by expressing the above operators in a "$$3\oplus 1$$ block form" as mentioned in the screenshot below. This might be something trivial or I might be missing something obvious but I fail to understand the transition from the $$4X4$$ form to the $$3X3$$ form. Really appreciate it if someone can point me in the right direction!! 