The number of purely real elements in a lower triangular representation of the given $3\times 3$ matrix, obtained through the given decomposition is ______________.

$$\begin{bmatrix} 2 &3 &3 \\ 3 &2 &1 \\ 3&1 & 7 \end{bmatrix}=\begin{bmatrix} a_{11} &0 &0 \\ a_{12}& a_{22} &0 \\ a_{13}&a_{23} &a_{33} \end{bmatrix}\begin{bmatrix} a_{11} &0 & 0\\ a_{12} &a_{22} &0 \\ a_{13}&a_{23} &a_{33} \end{bmatrix}^{\top}$$

1. $5$
2. $6$
3. $8$
4. $9$