Is 1,3 alkyl shift allowed in Cyclobutyldicyclopropylmethyl Cation? While reading a post , i came across the following mechanism .(source : Which ring expansion in cyclobutyl(cyclopropyl)methanol is favourable?)


The rearrangement is a series of shifts.


The first ring expansion is 1,2 alkyl shift.
The second ring expansion is 1,3 aalkyl shift.   
Third step is again 1,2 Hydride shift.


I am aware of 1,2 shift but not not 1,3 shift.So I researched from various sources which are tabulated below .


ORGANIC CHEMISTRY ( SECOND EDITION ) by Jonathan Clayden ,Nick Greeves, Stuart Warren .Quoting from page no. 942 "The methyl group can slide smoothly from one orbital to another—there are bonding interactions
all the way. The next step, migration of H, is just the same—except that the HOMO is now a C–H σ bond. 








in all the above , 1,2 shifts are seen.


Advanced Organic Chemistry , FIFTH EDITION ,Part A: Structure and Mechanisms
by Francis A. Carey and Richard J. Sundberg.

Quoting from page no.446 it says 

"The preferred alignment of orbitals for a 1,2-hydride or alkyl shift 
involves  coplanarity of the p-orbital at the carbocation ion center and the
orbital   of the migrating group "




1,2 shift pathway is seen.


https://www.masterorganicchemistry.com/2012/08/15/rearrangement-reactions-1-hydride-shifts/


Quoting from the above link.

"Here are some examples of “allowed” rearrangement reactions. Notice how we’re always going from a less substituted carbocation to a more substituted carbocation. One exception is at the very bottom; the rearrangement is favorable because the new carbocation is resonance stabilized."



A rearrangement can occur to give the more stable tertiary carbocation.(involves 1,2 shift)




https://www.masterorganicchemistry.com/2012/08/22/rearrangement-reactions-2-alkyl-shifts/


Quoting from the above link.The Mechanism Of Alkyl Migration Reaction is a 1,2 shift in two examples below.



Ring-Expansion Reactions




From https://en.wikipedia.org/wiki/Sigmatropic_reaction .This talks about Thermal alkyl [1,3] shifts.This type of mechanism is not possible for the rearrangement mechanism i am researching .The structure involved is totally different.




Therefor ,the above sources have all talked about 1, 2 shifts only . 

So my question is , is 1,3 alkyl shift as given in the post (Which ring expansion in cyclobutyl(cyclopropyl)methanol is favourable?)

  allowed ?
There may be 1,3-alkyl shifts but they are not needed here. Cation 1 should expand the more strained cyclopropane ring first to form cyclobutyl cation 2. After a 1,2-hydride shift to form a tertiary cation 3, the cyclobutane ring in 3 expands to form spirocyclopentyl cation 4. Another cyclobutane ring expansion of 4 leads to cation 5 and eventually to alkene 6.