Let $G$ be a Lie group; let $\alpha^X,\alpha^Y:\mathbb{R}\rightarrow G$ be a differentiable curves with $\alpha^X(0)=\alpha^Y(0)=e$ and let \begin{equation*} a(s,t):=\alpha^X(s)\cdot\alpha^Y(t)\cdot\alpha^X(-s). \end{equation*}Let $\varphi:G\rightarrow\mathbb{R}$ be differentiable.
How can I prove that \begin{equation*} \frac{\partial^2}{\partial s\partial t}\bigg|_0\varphi(a(s,t))=\frac{\partial^2}{\partial \sigma\partial t}\bigg|_0\varphi(\alpha^X(\sigma)\cdot\alpha^Y(t))-\frac{\partial^2}{\partial \tau\partial t}\bigg|_0\varphi(\alpha^Y(t)\cdot\alpha^X(\tau)) \end{equation*} using the chain rule for the composition $\varphi(a(s,t))=g(h(s,t))$ where \begin{align*} h:\mathbb{R}^2&\rightarrow\mathbb{R}^3\\ (s,t)&\mapsto(s,t,-s) \end{align*}and \begin{align*} g:\mathbb{R}^3&\rightarrow\mathbb{R}\\ (\sigma,t,\tau)&\mapsto\varphi(\alpha^X(\sigma)\cdot\alpha^Y(t)\cdot\alpha^X(\tau))? \end{align*}
If you are interested in the details about the context see Bröckner and Dieck's Representations of compact Lie groups, page 19.
Thank you!