Editing Lie algebroid (section)

=== First properties ===
It follows from the definition that

* for every <math>x \in M</math>, the kernel <math>\mathfrak{g}_x(A)=\ker(\rho_x)</math> is a Lie algebra, called the '''isotropy Lie algebra''' at <math>x</math>
*the kernel <math>\mathfrak{g}(A)=\ker(\rho)</math> is a (not necessarily locally trivial) bundle of Lie algebras, called the '''isotropy Lie algebra bundle'''
* the image <math>\mathrm{Im}(\rho) \subseteq TM</math> is a [[Singular distribution (differential geometry)|singular distribution]] which is integrable, i.e. its admits maximal immersed submanifolds <math>\mathcal O \subseteq M</math>, called the '''orbits''', satisfying <math>\mathrm{Im}(\rho_x) = T_x \mathcal{O}</math> for every <math>x \in \mathcal O</math>. Equivalently, orbits can be explicitly described as the sets of points which are joined by '''A-paths''', i.e. pairs <math>(a: I \to A, \gamma: I \to M)</math> of paths in <math>A</math> and in <math>M</math> such that <math>a(t) \in A_{\gamma(t)}</math> and <math>\rho (a(t)) = \gamma'(t)</math>
* the anchor map <math>\rho</math> descends to a map between sections <math>\rho: \Gamma(A) \rightarrow \mathfrak{X}(M)</math> which is a Lie algebra morphism, i.e.
:<math>\rho([X,Y])=[\rho(X),\rho(Y)] </math>

for all <math>X,Y \in \Gamma(A)</math>.

The property that <math>\rho</math> induces a Lie algebra morphism was taken as an axiom in the original definition of Lie algebroid.<ref name=":0" /> Such redundancy, despite being known from an algebraic point of view already before Pradine's definition,<ref>{{Cite journal|last=J. C.|first=Herz|date=1953|title=Pseudo-algèbres de Lie|journal=C. R. Acad. Sci. Paris|language=fr|volume=236|pages=1935–1937}}</ref> was noticed only much later.<ref>{{Cite journal|last1=Kosmann-Schwarzbach|first1=Yvette|last2=Magri|first2=Franco|date=1990|title=Poisson-Nijenhuis structures|url=http://www.numdam.org/item/AIHPA_1990__53_1_35_0/|journal=Annales de l'Institut Henri Poincaré A|volume=53|issue=1|pages=35–81}}</ref><ref>{{Cite journal|last=Grabowski|first=Janusz|date=2003-12-01|title=Quasi-derivations and QD-algebroids|url=https://www.sciencedirect.com/science/article/pii/S0034487703800411|journal=Reports on Mathematical Physics|language=en|volume=52|issue=3|pages=445–451|doi=10.1016/S0034-4877(03)80041-1|issn=0034-4877|arxiv=math/0301234|bibcode=2003RpMP...52..445G|s2cid=119580956}}</ref>