In section 4 of this paper the authors use a given list of Clebsch-Gordan coefficents for the $27 \otimes 27$ of $E_6$ from an old paper and combine it with their own list of Clebsch-Gordan coefficents for $27 \otimes \bar{27}$ of $E_6$ to compute the Clebsch-Gordan coefficents for $27\otimes 27 \otimes 27$. Unfortunately, they don't give any details how this is done.
The only hint they offer is that there is a $\bar{27}$ in $27 \otimes 27$, i.e.
$$ 27 \otimes 27 = \bar{27} \oplus \ldots $$
Therefore
$$ 27\otimes 27 \otimes 27 = ( \bar{27} \oplus \ldots ) 27 = 1 \oplus \ldots .$$
In fact, the authors (and I) are only interested how the $1$ decomposes under $ 27\otimes 27 \otimes 27 $.
From the authors list of Clebsch-Gordan coefficients we know how the $1$ decomposes under $27 \otimes \bar{27}$. How can this knowledge be combined with a list of Clebsch-Gordan coefficients for the $\bar{27}$ under $ 27\otimes 27 $ to yield how the $1$ decomposes under $ 27\otimes 27 \otimes 27 $?
Any ideas or tips how this can be done would be much appreciated!