On matrix decomposition of one reduced cubic irrational

In this work we considered the matrix decomposition of the cubic irrational $\alpha$ satisfying the equation
$$x^3 - 4x^2 - 5x - 1 = 0.$$

For decomposition of the matrix
$$ \left(
\begin{array}{c} \alpha \\ 1 \\ \end{array}
\right)=\prod_{k=0}^\infty\left(
\begin{array}{cc} 310941\cdot k+155427 & 156744\cdot k+78333 \\ 61578\cdot k+30882 & 31041\cdot k+15564\\ \end{array}
\right) $$
an algorithm of transition to regular continued fraction is constructed.
Bibliography: 2 titles.