Manticoran Peerage seating
All of the peerages above the rank of Baron are represented by two seats: the seat for the current holder of the title, and the "cadet" seat for the same peerage (held by the heir). Thus, when we first meet the Alexander Brothers, their father is Earl White Haven and holds the White Haven seat; Hamish, as his eldest child, is "Lord Alexander," and holds the White Haven cadet seat; and Willie is "the Honorable William" (indicating he is the child of a peer, but not the peer's heir). When Hamish succeeded to the title, he became Earl White Haven, and Willie became "Lord Alexander." For another example, look at Mike Henke and her older brother. Mike is "the Honorable Michelle," her brother is "Lord Henke," and her father is Earl Gold Peak. (Her mother, as the Queen's aunt, is a special case.) There are instances in which two seats (and votes) are combined in a single person, as for example when a single peer holds dual titles and has no heir. The Star Kingdom of Manticore's constitution precludes any individual's holding more than two seats (and votes). In the very rare instances in which three or more titles are united in a single individual, only the two most senior of the titles are allowed seats/votes. For example, if Honor had retained her title as Countess Harrington, in theory she could have been ennobled a second time as Duchess Harrington, in which case she would have held two seats: one as countess and one as duchess. Moreover, she would have been entitled to two cadet seats, one for each title, although under normal circumstances only one cadet seat would have been filled because the same person would have been heir to both titles. (That is, before the birth of her sister and brother, her cousin Devon should technically have been seated in the Lords as "Lord Harrington." The... irregularity of Honor's title and the fact that she was denied her seat in her own right as Countess Harrington after her duel with Pavel -- which occurred before Devon would normally have been seated, anyway -- in effect destroyed the Harrington seat in the House of Lords entirely, and with it the cadet seat. When Devon inherited her title as Earl Harrington, the situation "reset," and his son is now "Lord Harington" whereas Faith Harrington, in addition to being Honor's Grayson heir, is also "Lady Harrington," and would be entitled to the cadet seat of the Duchy of Harrington upon reaching her legal majority. In addition, the Crown may, on its own authority, create as many nonvoting life-peers as it likes. Those peerages are not inheritable and their holders are not entitled to seats in the House of Lords at all. The Crown and Commons may -- with the consent of a majority of the House of Lords -- jointly create voting life peers. The title is not inheritable, and, as such, carries no cadet seat, but the holder of such a peerage is a voting member of the House of Lords for the remainder of his life and may be denied his vote only if the Lords later vote to exclude or the Commons votes to sustain an impeachment.
Does that sufficiently confuse the issue?
The only member of a cabinet who must come from the Lords is the PM him/herself. The other ministers technically need not even be a member of Parliament, although it is traditional that all cabinet-level ministers are drawn from one House or the other. Willie could not have succeeded Cromarty in the premiership before Hamish succeeded his father, thus elevating Willie to the cadet White Haven seat. Of course, under normal circumstances, Elizabeth would have nominated Willie for a peerage (probably a life peerage, without right of inheritance) after Cromarty's death, and the Commons would undoubtedly have confirmed, given the Centrists' majority there, which would have entitled Willie to form a government. The point was moot by the time Cromarty actually did die, because Willie had been sitting in the Lords by years at that point.
No life peerage carries a cadet seat with it, but a very small handful or particularly old baronies do carry cadet seats, and the Queen and Commons have the right to create additional baronies which carry cadet seats, although they haven't done so in the past couple of centuries.