Motivation for the definition "strongly real element" in group? Elements in finite groups can splitted in real / non-real elements. 
Which is quite well-motivated definition: 
element is called real if all characters take real values on it.
Equivalent requirment is:  element is conjugate to its inverse. 
Definition: There is stronger property: element is called strongly real
 if it satisfies the following equivalent conditions:


*

*It is either the identity element or an involution or can be expressed as a product of two distinct involutions (here an involution means a non-identity element whose square is the identity element).

*It is either the identity element or there is an involution that conjugates it to its inverse.


Question: What is the motivation of that definition  ?

Further background:
It is known that the number of real conjugacy classes is equal to the number
of the real characters. 
Real characters can be splitted further by e.g. Frobenius-Shur indicator
as real/quaternionic characters.
However such splitting of characters is NOT reflected by splitting conjugacy classes to strongly real. See e.g. MO Are there “real” vs. “quaternionic” conjugacy classes in finite groups?, MO If all real conjugacy classes are strongly real, then all real irreps are “strongly real”(symmetric), true ?,
MO Strongly real elements of odd order in sporadic finite simple groups
Remarks: Real and strongly real elements in finite simple groups 
have been much studied see e.g. presentation Singh,  arXiv:1104.3933,
arXiv:0809.4412 , arXiv:1303.6085
 A: The real elements of $G$ are exactly those elements $x$ in $G$ such that
$x^g = x^{-1}$ for some element $g \in G$. Also, $x$ is strongly real if
it is real and we can find $g$ as above with the additional property that
$g^2 = 1$. Thus "strongly real" is a more stringent requirement than merely
being real.
The condition that $x$ is real is equivalent to saying that $\chi(x)$ is a real number for all characters $\chi$ of $G$. The above definition of "strongly real" is equivalent to the definition given in the question, but is, I think, cleaner and more concise. The property of being strongly real is not reflected in the character values. Note that $D_8$ and $Q_8$ have identical character tables, but in $D_8$, every element is strongly real, and in $Q_8$, only two elements are strongly real.
A: Strongly real elements are important in several counting theorems,and play an important role in the famous Brauer-:Fowler Annals paper "On groups of even order", ( circa 1955-56) which may be where they are first defined (though I have not checked).
