Let $(G\rightrightarrows M)$ be a Lie groupoid (i.e. a groupoid with source map $s$ and target map $t$ such that $G,M$ are smooth manifolds and the structural maps are all smooth (and $s$,$t$ are submersions).

Defining $$IG := \{g \in G \mid s(g) = t(g) \}$$ we obtain the so called isotropy subgroupoid $IG \rightrightarrows M$ which is in general only a subgroupoid but not a sub LIE groupoid, in that $IG$ need not be a submanifold of $G$ (for an example consider the action groupoid of the canonical action $\mathbb{S}^1 \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$ of the circle group via rotation. In this case $G=\mathbb{S}^1 \times \mathbb{R}^2$ and thus $IG = \mathbb{S}^1 \sqcup \bigsqcup_{x\in \mathbb{R}^2} \{1\}$, whence the isotropy subgroupoid of the action groupoid can not be a Lie groupoid with respect to the subspace topologies.

Now I have read in several places (cf. e.g. 1) that the isotropy groupoid is an embedded Lie subgroupoid if the Lie groupoid is regular. Recall that a Lie groupoid is regular if its anchor $$(s,t)\colon G \rightarrow M\times M, g \mapsto (s(g),t(g))$$ is a mapping of constant rank (there are some other equivalent formulations, using e.g. the orbit foliation of the Lie groupoid, see e.g. 2). Unfortunately, I was not able to track down a proof of this folklore fact in the literature.

The old book by Mackenzie (Lie Groupoids and Lie Algebroids in Differential Geometry) proves a similar statement using foliation charts for the orbit foliation. Note that the statement in the book claims that it is true for all Lie groupoids (there called differentiable groupoids), but the proof holds only for locally trivial Lie groupoids (and seems to be intended to be only for this class). For this class however, the anchor $(s,t)$ is a submersion (and $IG = (s,t)^{-1} (\Delta M)$, where $\Delta M$ is the diagonal) whence it is easy to see why $IG$ is an (embedded) submanifold of $G$.

I wonder now how one tackles the general case for regular Lie groupoids. Surely there must be a proof somewhere in the literature? Or alternatively an easy argument which eludes me?