& \lambda^2_0 & \lambda_0 & -\lambda^0_0 \\
                                 \end{array}
                               \right).$$ The entries $$\begin{split}
& v,m,\lambda^0_0,\lambda^1_0,\lambda^2_0,\lambda_{n+1}^1,\lambda^2_{n+1}
:I\times (-\epsilon,\epsilon)\to \R,\\
& p, \lambda_0,\lambda_2,\lambda_2,\lambda_{n+1}:I\times (-\epsilon,\epsilon)\to \R^{n-2},\\
& \psi,\lambda :I\times (-\epsilon,\epsilon)\to \mathfrak{o}(n-2).
\end{split}$$ of $Q$ and $\Lambda$ are smooth maps. Their restrictions to $I\cong I\times \{0\}$ are denoted with the same symbols with an over-bar. Note that $\mathrm{supp}(\overline{\Lambda})\subseteq K$. The cross section $$V_{\mathbf{g}}=\sum_{a=2}^{n}\overline{\lambda}^a_0 A_a:I\to \mathcal{N}^2(\gamma)$$ does not depend on the choice of $A$ and $\mathcal{A}$. We call $V_{\mathbf{g}}$ the [*infinitesimal variation*]{} of $\mathbf{g}$. From $\dot{A}=A\cdot \overline{Q}$ and taking into account (\[LS\]), we obtain $\overline{\upsilon}=1$, $\overline{m}_1=h_1$, $\overline{p}=s$ and $\overline{\psi}=\phi$. From the Maurer-Cartan equations we have $$\partial_u\Lambda - \partial_tQ = \Lambda\cdot Q - Q\cdot \Lambda.$$ This implies $$\begin{split}
&(\partial_t v)|_I = \overline{\lambda}_0^0 + \dot{\overline{\lambda}}^{1}_{0},\\
&\overline{\lambda}_0^0 =\frac{1}{2}(\dot{\overline{\lambda}}^0_{2}-\dot{\overline{\lambda}}^1_{0}-
\overline{\lambda}_{n+1}\cdot s-h_1\overline{\lambda}^2_1),\\
&\dot{\overline{\lambda}}_{0}=\overline{\lambda}_1+\overline{\lambda}_0^2s-\phi \cdot \overline{\lambda}_0,\\
&\dot{\overline{\lambda}}_1=-h_1\overline{\lambda}_0-\overline{\lambda}_{n+1}-\overline{\lambda}^2_1s-\phi\cdot\overline{\lambda}_1,\\
& \dot{\overline{\lambda}}_{n+1}=h_1\overline{\lambda}_1+\overline{\lambda}_2-\overline{\lambda}^2_0s-\phi\cdot \overline{\lambda}_{n+1},\\
&\dot{\overline{\lambda}}_0^2=\overline{\lambda}^2_1-s\cdot \overline{\lambda}_0.
\end{split}$$ From these equalities, integrating by parts and taking into account that $\mathrm{supp}(\overline{\lambda}^i_j)\subset K$, $i,j=0,\dots, n+1$, we have $$\begin{split}
&\frac{d}{dt}\left(\mathcal{S}_K(\gamma_t)\right)|_{t=0}=\int_K (\partial_t v|_I)du =
\int_K (\overline{\lambda}_0^0+\dot{\overline{\lambda}}^1_0)du=\int_K \overline{\lambda}_0^0du=\\
&= -\frac{1}{2}\int_K (s\cdot(-\dot{\overline{\lambda}}_1-
h_1\overline{\lambda}_0-\overline{\lambda}_1^2 s-\phi\cdot \overline{\lambda}_1 )+h_1(\dot{\overline{\lambda}}_0^2+s\cdot \overline{\lambda}_0))du=\\
&=\frac{1}{2}\int_K(-\dot{s}\cdot\overline{\lambda}_1+h_1s\cdot \overline{\lambda}_0+(s\cdot s)\overline{\lambda}^2_1+s\cdot \phi\cdot \overline{\lambda}_1+\dot{h}_1\overline{\lambda}^2_0+h_1s\cdot\overline{\lambda}_0)du=\\
&=-\frac{1}{2}\int_K(\dot{s}\cdot(\dot{\overline{\lambda}}_0+\overline{\lambda}_0^2s+\phi\cdot \overline{\lambda}_0))du+\frac{1}{2}\int_K(s\cdot \phi\cdot (\dot{\overline{\lambda}}_0+\overline{\lambda}^2_0 s+\phi\cdot \overline{\lambda}_0))du\\
&+\frac{1}{2}\int_K(k_1s\cdot \overline{\lambda}_0+(^ts\cdot s)(\dot{\overline{\lambda}}^2_0-s\cdot \overline{\lambda}_0)+\dot{h}_1\overline{\lambda}^2_0+h_1s\cdot \overline{\lambda}_0)du=\\
&=\frac{1}{2}\int_K(\overline{\lambda}^2_0(\dot{h}_1-3 ^ts\cdot \dot{s})+
\overline{\lambda}_0\cdot (\dot{s}_{(1)}+\phi\cdot s_{(1)}-((^ts\cdot s)-2h_1)s))du=\\
&=\frac{1}{2}\int_K(\overline{\lambda}^2_0(\dot{h}_1-3 ^ts\cdot \dot{s})+
\overline{\lambda}_0\cdot (s_{(2)}+((2h_1-^ts\cdot s))s))du= \frac{1}{2}\int_K \langle V_{\mathbf{g}},W_{\gamma}\rangle du.
\end{split}$$ This proves 