Two faces of $N=7,8$ superconformal mechanics

Two variants of $N=7$ superconformal mechanics with manifest $so(7)$ and $su(2)\times su(2)$ $R$-symmetries possessing exceptional $G(3)$ dynamical symmetry are presented. To construct these $G(3)$ theories, we developed two new versions of covariant embedding of the exceptional $g_2$ algebra into the $so(7)$ algebra and then the embedding of $so(7)$ into $so(8)$. As a result, two $N=8$ superconformal mechanics with the $OSp(8\mathrm{\mid }2)$ and $F(4)$ superconformal algebras and $N=7,G(3)$ superconformal mechanics were constructed in a uniform way. The constants of the octonion multiplication play a key role in the construction of superconformal mechanics with manifest $so(7)$ symmetry.