Let $n$ be a positive integer. Denote $ \left[ n \right] \equiv \{1, \ldots , n \}$. Denote $ T_{n} \left( x \right) $ as the $n$-th Chebyshev polynomials of the first kind. Let $ m_{1}, \ldots , m_{n} $ be positive integers. Then an "index-decomposition" identity for the Chebyshev polynomial of the first kind is given by $$ \begin{align} T_{m_{1} + \dots + m_{n}} \left( x \right) & = \sum_{\begin{aligned} I \subseteq \left[ n \right] \, \\ \lvert I \rvert \, \text{even} \end{aligned}}{ \left( -1 \right)^{\frac{\lvert I \rvert}{2}} \prod_{i \in I}{\sqrt{1-T_{m_{i}} \left( x \right)^{2}}} \prod_{j \in \left[ n\right] \setminus I}{T_{m_{j}} \left( x \right)}} \\ & = \sum_{\begin{aligned} I \subseteq \left[ n \right] \, \\ \lvert I \rvert \, \text{even} \end{aligned}}{ \left( \frac{T_{2} \left( x \right) -1}{2} \right)^{\frac{\lvert I \rvert}{2}} \prod_{i \in I}{ \frac{T_{m_{i}} \left( x \right)'}{m_{i}}} \prod_{j \in \left[ n\right] \setminus I}{T_{m_{j}} \left( x \right)}} \end{align} $$ This identity came up in the context of a previous question.
I am asking for references to the identity; preferably those which also consist of applications and/or generalizations.
