Does the kinship and inbreeding coefficients depend on population frequency of an allele?

I am reading Section 5.2, Kinship and Inbreeding Coefficients, of Kenneth Lange, Mathematical and Statistical Methods for Genetic Analysis. There the kinship coefficient $Phi_{i,j}$ is defined for two relatives $i$ and $j$ as the probability that a gene selected randomly from $i$ and a gene selected randomly from the same autosomal locus of $j$ are identical by descent.

I would think $Phi_{i,j}$ should depend on the particular population frequency of the gene or allele at that locus. However, the book does not seem to indicate that or at least does not stress that at all. Is there a dependence or not? The same question applies to the inbreeding coefficient.

More importantly, I doubt this definition is mathematically rigorous.

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Here is an example to make what puzzles me clearer. It must be that the probability I am computing is not what the kinship coefficient. But this is the definition means to me. Some expert please elucidate the correct meaning of the definition.

Consider the simplest pedigree tree of one family of two parents with brothers $A$ and $B$. Allele $a$ is detected at a particular locus. Let $b$ be the other allele of the same gene. We ask for the probability of the brother $B$ having allele $a$ conditioned on $A$ having allele $a$.

In the following, to clutter symbols, we abuse the symbols by use $a$ to denote the population frequency or the unconditional probability of finding allele $a$ at the locus for the whole population as $a$. The same goes for allele $b$. $a,bin[0,1]$ and $a+b=1$. We solve this problem by listing as follows all possible parental gene configurations, the configuration probability (considering the symmetry of paternal and maternal loci), as well as the probability finding allele $a$ for either $A$ or $B$ at the locus.

begin{align}
text{parent config}~~~~ & text{Pr(config)} & text{Pr($a$|config)}
aa|aa~~~~~~~~~ & ~~a^4 & 1
aa|ab~~~~~~~~~ & 4a^3b & 1
aa|bb~~~~~~~~~ & 2a^2b^2 & 1
ab|ab~~~~~~~~~ & 4a^2b^2 & frac34
ab|bb~~~~~~~~~ & 4ab^3 & frac12
bb|bb~~~~~~~~~ & ~~~~b^4 & 0
end{align}

The joint conditional probability of both $A$ and $B$ have allele $a$ at the locus is Pr($a$|config)$^2$. So the required conditional probability is
$$P(r):=text{Pr}(Btext{ has } a|Atext{ has }a)=frac{sum_text{config} text{Pr(config)Pr(a|config)}^2}{sum_text{config} text{Pr(config)Pr(a|config)}} = frac{r^3+4r^2+2r+4r(frac34)^2+frac4{2^2}}{r^3+4r^2+2r+4rfrac34+frac42}, quad r:=frac abin [0,infty).$$
$P(r)$ depends on $r$ which depends on the population frequency of the alleles as I have claimed.

Consider $[r^3+4r^2,2r,3r, 2]$ as a positive entry vector. $P$ is a convex combination of $v:=big[1,1,1,frac34,frac12big]$. So $P$ is between the minimum and maximum of the entries of $v$, and
$$frac12le Ple 1.$$
We see the extrema are achieved at
$$P(r)=begin{cases}
frac12, quad r=0 [0.7ex]
1,quad rtoinfty
end{cases}$$