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Cartan-Lie algebroids, i.e. Lie algebroids equipped with a compatibleconnection, permit the definition of an adjoint representation, on the fiber aswell as on the tangent of the base. We call (positive) quadratic Liealgebroids, Cartan-Lie algebroids with ad-invariant (Riemannian) metrics ontheir fibers and base $\kappa$ and $g$, respectively. We determine thenecessary and sufficient conditions for a positive quadratic Lie algebroid tointegrate to a Riemmanian Cartan-Lie groupoid. Here we mean a Cartan-Liegroupoid $\mathcal{G}$ equipped with a bi-invariant and inversion invariantmetric $\eta$ on $T\mathcal{G}$ such that it induces by submersion the metric$g$ on its base and its restriction to the $t$-fibers coincides with $\kappa$.

Integration of quadratic Lie algebroids to Riemannian Cartan-Lie groupoids