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In his 1910 "Five Variables" paper, Cartan solved the equivalence problem forthe geometry of $(2, 3, 5)$ distributions and in doing so demonstrated anintimate link between this geometry and the exceptional simple Lie groups oftype $\textrm{G}_2$. He claimed to produce a local classification of all such(complex) distributions which have infinitesimal symmetry algebra of dimensionat least $6$ (and which satisfy a natural uniformity condition), but in 2013Doubrov and Govorov showed that this classification misses a particulardistribution $\bf E$. We produce a closed form for the Fefferman-Graham ambientmetric $\widetilde{g}_{\bf E}$ of the conformal class induced by (a real formof) $\bf E$, expanding the small catalogue of known explicit, closed-formambient metrics. We show that the holonomy group of $\widetilde{g}_{\bf E}$ isthe exceptional group $\textrm{G}_2^*$ and use that metric to give explicitly aprojective structure with normal projective holonomy equal to that group.  We also present some simple but apparently novel observations about ambientmetrics of general left-invariant conformal structures that were used in thedetermination of the explicit formula for $\widetilde{g}_{\bf E}$.

Cartan's incomplete classification and an explicit ambient metric of holonomy $\mathrm{G}_2^*$