First time appearance of Lie crossed module (crossed module of Lie groups) in literature Can someone point me to a reference where the notion of "Lie crossed module" appeared for the first time?
I see many papers "recall" the definition of the Lie crossed module but, I do not see any mention of a "first-time" reference.
The definition of Lie crossed module I am referring to is mentioned as Definition 1.3  in

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*João Faria Martins and Roger Picken, On Two-Dimensional Holonomy, Trans. Amer. Math. Soc. 362 (2010) pp. 5657-5695, doi:10.1090/S0002-9947-2010-04857-3, arXiv:0710.4310.

 A: Crossed modules of Lie algebras are defined by Kassel and Loday in Definition A.1 of Extensions centrales d’algèbres de Lie (published 1982).
Crossed modules of Lie groups are defined by Mackenzie in Definition 1.5 of Classification of principal bundles and lie groupoids with prescribed gauge group bundle.
Definition 3.3 there defines crossed modules of Lie groupoids (published 1989).
Crossed modules of local Lie groupoids are defined by Brown and İçen in Definition 4.1 of Towards a 2-dimensional notion of holonomy (published 2003).
A: For some more context, Kassel and Loday's 1982 paper defining crossed modules of Lie algebras cites an early paper by Loday that discusses crossed modules of (ordinary) groups, so it would appear that this citation trail won't give you crossed modules of Lie groups. Elsewhere I've seen crossed modules mentioned in conjunction with "abstract kernels", which date back to Eilenberg and Mac Lane's work on classifying nonabelian extensions of groups (which is secretly controlled by non-abelian cohomology, and the explicit group cohomology $H^3$—with values in an abelian group—that they use also helps classify 2-groups):  Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel.
Mackenzie in his 1989 paper defining crossed modules of Lie groups cites a 1960 paper on analytic abstract kernels:

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*R. A. Macauley, Analytic group kernels and Lie algebra kernels, Trans. Amer. Math. Soc. 95 (1960) pp. 530-553, doi:10.1090/S0002-9947-1960-0122908-6
Analytic groups here can be thought of as at least a special case of Lie groups (at the very least, classical groups are analytic), and the Macauley paper deals with constructions that turn up in the study of crossed modules of Lie groups, like induced maps to outer automorphism groups. So one could say that this latter paper almost invented crossed modules of Lie groups. But it's not too surprising that it doesn't, since even Eilenberg and Mac Lane's Annals paper involving abstract kernels (from 1947) don't cite Whitehead, even though they are even closer in subject matter (Mac Lane and Whitehead collaborated later in 1950, producing On the 3-type of a complex, and there crossed modules turn up). I think it unlikely someone working with Lie groups would have been familiar with work in algebraic homotopy theory or algebraic topology so as to even know of Whitehead's and/or Mac Lane–Whitehead's work.
If Mackenzie, who was an early proponent of Lie groupoids, could only cite someone who is a near-miss for crossed modules of Lie groups, then it's safe to say his paper is probably the earliest. The only others I could have imagined independently developing the notion would be Jean Pradines (another Lie groupoidist, from the Ehresmann school) or perhaps someone in Ronnie Brown's orbit arriving at the idea of a topological crossed module via an analogue of the Brown–Spencer theorem relating groupoids internal to $\mathbf{Grp}$ and crossed modules), but I briefly checked for the latter and couldn't find anything.
