PER CURIAM:

Lawrence B. Cummings ("Lawrence"), proceeding pro se, appeals from the district court's affirmance of the bankruptcy court's orders determining that his obligations under a state court divorce decree are partially dischargeable and partially non-dischargeable, and denying recusal of the bankruptcy judge. On appeal, Lawrence argues that (1) the bankruptcy court erred in finding that it was barred by the "law of the case" doctrine from independently redetermining what portion, if any, of the underlying state divorce court judgment was intended to be in the nature of "support" within the meaning of 11 U.S.C. § 523(a)(5); and (2) the bankruptcy judge abused his discretion in denying Lawrence's motion for recusal. Upon review of the record and the parties' briefs, we discern no reversible error and AFFIRM the judgment of the district court.

I. BACKGROUND

Lawrence and Susan Cummings ("Susan") were divorced in 1996. The state divorce court awarded Susan $6,300,000 as an "equitable distribution," payable in three annual lump sum installments of $2,100,000 each, but did not indicate what portion of the award, if any, was intended as support. Cummings v. Cummings, 244 F.3d 1263, 1264-65 (11th Cir. 2001) (" Cummings I"). Shortly before the first installment was due, Lawrence filed a bankruptcy petition seeking to discharge his obligation to Susan under the divorce decree. The bankruptcy court initially found that the entire award was dischargeable as a property settlement, but we vacated that order discharging the award on appeal. Cummings I, at 1267. In particular, we noted that the state divorce court had concurrent jurisdiction with the bankruptcy court to determine whether the divorce obligation was "in the nature of support" for purposes of § 523(a)(5), and we ruled that the bankruptcy court should await the divorce court's clarification of its intent regarding the amount of support provided in its award. Id.

On remand, however, the bankruptcy court entered an order finding that it had exclusive jurisdiction over the case and enjoining Susan from seeking a clarification order from the divorce court. R1-26 at 3. Susan petitioned us for a writ of mandamus, which we granted, directing the bankruptcy court (1) to vacate its "[e]xclusive [j]urisdiction order," (2) not to hinder Susan "from obtaining a clarification from the state divorce court as to what portion, if any, of the equitable distribution was intended as support," and (3) "not to schedule further proceedings on the dischargeability issue until [Susan] had had a reasonable opportunity to obtain clarification from the [divorce] court". Id. at 3-4. The divorce court then entered an order clarifying that it intended the first two lump sum payments to serve as support, but not the third payment. R1-26 at 4. Lawrence appealed the merits of that order, and the state appellate court affirmed without opinion the divorce court's order. Cummings v. Cummings, 835 So.2d 1143 (Fla. 4th Dist.Ct.App. 2002) (table).

Thereafter, based on the divorce court's clarification order, the bankruptcy court found that, the first two payments were nondischargeable under § 523(a)(5), while the third payment was dischargeable. The bankruptcy court refused to consider Lawrence's attempt to challenge the divorce court's finding that the first two payments were intended to be in the nature of support, finding that the issue was barred by the "law of the case" doctrine. Accordion Folder 1-11-4 at 8. Lawrence appealed the bankruptcy court's ruling, and the district court affirmed.

Meanwhile, Lawrence also had unsuccessfully moved to recuse the bankruptcy judge for bias, based on two specific statements. The first statement appeared within a footnote of an order denying Susan's emergency motion for financial support; the bankruptcy judge compared Lawrence's comfortable lifestyle, based on his apparent ability to pay for the services of a prestigious South Florida law firm, to Susan's "deplorable" living conditions and pro se status. R1-26 at 4-5. The second statement was made during a 2001 hearing in Susan's separate bankruptcy proceedings upon the confirmation of Susan's own bankruptcy reorganization plan; the bankruptcy judge expressed "hope that Ms. Cummings is successful in her pursuit as far as the recovery of moneys owed to her. I sincerely mean that." Id. at 5. The bankruptcy court denied the motion for recusal, and the district court held that the bankruptcy court did not abuse its discretion in denying the motion because the two challenged comments (1) did not stem from an extrajudicial source, (2) were "only two fairly innocuous statements" made over the course of approximately ten years of legal proceedings, and (3) would not lead "any reasonably objective lay person" to doubt the bankruptcy judge's impartiality. Id. at 15-16. Lawrence, who had been represented by counsel throughout the bankruptcy and district court proceedings, filed a timely appeal pro se.

II. DISCUSSION

First, Lawrence argues that, on remand following Cummings I, neither the bankruptcy court nor the divorce court followed our "clear guidelines" to determine the original intent of the divorce court as to what portion of the divorce judgment should function as support. Appellant's Br. at 7. Lawrence also asserts that the first two lump sum installments total more than $4,000,000, well over the $8,000 per month that the divorce court found that Susan needed to maintain her accustomed standard of living. In light of that discrepancy, Lawrence maintains that the divorce court's clarification that the first two of the three lump sum payments were intended as support was arbitrary. Id. In his reply brief, Lawrence explains that he is not trying to "re-litigate" any issue, but rather, he wants "a court" to consider for the first time his evidence that the divorce court's 2002 clarification order did not accurately reflect that court's original intent in its 1996 divorce decree. Reply Br. at 1.

We have "jurisdiction over all final orders of a district court exercising appellate jurisdiction over bankruptcy court orders." In re Pugh, 158 F.3d 530, 532 (11th Cir. 1998), (citing 28 U.S.C. § 158(d)). "In exercising such jurisdiction, we review conclusions of law made by the district and bankruptcy courts de novo." Id. Claim preclusion applies to an order or judgment when four conditions are met: (1) "the prior judgment must be valid in that it was rendered by a court of competent jurisdiction and in accordance with the requirements of due process"; (2) "the judgment must be final and on the merits"; (3) "there must be identity of both parties or their privies"; and (4) "the later proceeding must involve the same cause of action as involved in the earlier proceeding." In re Justice Oaks II, Ltd., 898 F.2d 1544, 1550 (11th Cir. 1990). By contrast, under the closely related doctrine of the "law of the case," "when a court decides upon a rule of law, that decision should continue to govern the same issues of subsequent stages of the same case." Christianson v. Colt Indus. Operating Corp., 486 U.S. 800, 815-16, 108 S.Ct. 2166, 2177, 100 L.Ed.2d 811 (1988) (citation omitted). We have explained the difference between claim preclusion and law of the case as follows:

[w]hile the law of the case does not bar litigation of issues `which might have been decided but were not[.]' [I]t does require a court to follow what has been decided explicitly, as well as by necessary implication, in an earlier proceeding. The distinction between law of the case and claim preclusion is that the former bars relitigation of legal rules while the latter bars relitigation of claims (i.e., legal rules applied to the facts of the case). In addition, law of the case bars only those legal issues that were actually, or by necessary implication, decided in the former proceeding, while claim preclusion bars relitigation not only of claims raised but also claims that could have been raised.

Justice Oaks II, 898 F.2d at 1549 n. 3 (citations omitted).

Here, the bankruptcy court correctly found that, in light of the divorce court's clarification order, it was precluded by the "law of the case" doctrine from re-litigating the issue of to what extent the divorce court intended its award to constitute support. In Cummings I, we expressly acknowledged the state court's authority and concurrent jurisdiction to determine whether an obligation is in the nature of support within the meaning of § 523(a)(5). Cummings I, 244 F.3d at 1267. We stressed that bankruptcy courts should avoid "incursions into family law matters out of consideration of court economy, judicial restraint, and our deference to our state court brethren and their established expertise in such matters." Id. When the bankruptcy court on remand thereafter sought to assert exclusive jurisdiction over the case and enjoin Susan from seeking the divorce court's clarification, we granted mandamus relief and expressly directed the bankruptcy court to vacate its "[e]xclusive [j]urisdiction order" and abate further proceedings on the dischargeability issue until Susan had the opportunity to obtain such clarification. R1-26 at 3-4. As such, our two decisions established the rule that the bankruptcy court's review was limited only to determining the intent of the divorce court when it made the award, not second guessing the amount of the award that should have been labeled as support. Thus, the bankruptcy court properly disregarded Lawrence's attempt to present additional evidence to show that the award amount was excessive when compared to Susan's monthly support needs.

As we noted in Cummings I, the divorce court was a court of competent jurisdiction to consider the support issue, and neither party disputes that the divorce court's decision met the requirements of due process. The divorce court's clarification order resolved the support issue on the merits, Lawrence appealed that order, and the state appellate court affirmed. The parties were identical in both proceedings. Thus, once the divorce court resolved the issue of its intent, Lawrence was precluded from relitigating that claim before the bankruptcy court. See Justice Oaks II, 898 F.2d at 1550. At that point, the only thing left for the bankruptcy court to do was to take the final analytical step and apply the divorce court's factual clarification to the dischargeability context under § 523(a)(5). Notably, our decision in Cummings I also spoke to that issue as well, clarifying that "[t]o the extent the [divorce] court intended a portion of the obligation to function as support, that debt is nondischargeable under § 523(a)(5)." Cummings I, 244 F.3d at 1266.

The bankruptcy court correctly found that it was barred by the "law of the case" doctrine from independently determining to what extent the divorce court's judgment was intended to be in the nature of support within the meaning of § 523(a)(5). Accordingly, the bankruptcy court properly found that the first two lump sum payments were not dischargeable under § 523(a)(5) based on the divorce court's clarification order.

Second, Lawrence contends that the bankruptcy judge abused his discretion in failing to recuse himself from the adversary proceeding. In support of his motion, Lawrence argues that the bankruptcy judge's comment regarding Lawrence's ability to pay for the services of a prestigious South Florida law firm while Susan and the Cummings's children "suffer[ed] from deplorable living conditions" was directly refuted by overwhelming evidence in the bankruptcy case establishing his own "dire financial condition," and thus must have been developed from an extrajudicial source of information. R1-26 at 4-5. In addition, Lawrence points to the bankruptcy judge's on-the-record statement that he "hope[d] that Ms. Cummings is successful in her pursuit as far as the recovery of moneys owed to her," which Lawrence labels as "cheerleading" and evidence of an "obvious bias." R1-24 at 13-15.

We review the denial of a motion for recusal for abuse of discretion. Christo v. Padgett, 223 F.3d 1324, 1333 (11th Cir. 2000). The proper standard under § 455(a) is "whether an objective, fully informed lay observer would entertain significant doubt about the judge's impartiality." Id. A judge must be recused if the judge is personally biased or prejudiced against a party or in favor of an adverse party, or whenever the judge's "impartiality might reasonably be questioned." 28 U.S.C. §§ 144, 455(a). "The exception to this rule is when a judge's remarks in a judicial context demonstrate such pervasive bias and prejudice that it constitutes bias against a party." Id. (internal quotations omitted).

In this case, the bankruptcy judge did not abuse his discretion in denying Lawrence's motion for recusal. The only grounds cited in support of the motion were two isolated comments made over the course of ten years of protracted litigation. Despite Lawrence's suggestion to the contrary, neither of the two challenged comments was based on information derived from an extrajudicial source. First, the bankruptcy judge's juxtaposition of Lawrence's apparent ability to afford the services of a prestigious law firm to Susan's "deplorable" living conditions and pro se status in that court was a comment based on the judge's firsthand knowledge of the bankruptcy pleadings themselves. R1-26 at 5. That comment directly related to an issue then before the court, namely, Susan's asserted entitlement to emergency financial support, a request that the court coincidentally denied. Lawrence's contention that the judge necessarily must have consulted some unidentified extrajudicial source of information, while disregarding the record evidence of Lawrence's own "dire financial condition," is unsupported and baseless. R1-24 at 14. Second, the bankruptcy judge's comment regarding his hope that Susan would be successful in collecting money owed to her also stemmed solely from a judicial source; it occurred on the occasion of the confirmation of her own bankruptcy reorganization plan. Contrary to Lawrence's claim that the judge was cheerleading Susan's attempts to collect the equitable distribution from Lawrence, we agree with the district court that the bankruptcy judge reasonably could have made this statement as an innocuous expression of hope in her ability to one day satisfy her individual debts to her own creditors. See R1-26 at 14.

In light of the extensive history of this case, we conclude that the two isolated comments Lawrence relies upon do not evince pervasive bias and prejudice against Lawrence, and we determine that a reasonable, fully informed objective observer would not entertain any significant doubt about the bankruptcy judge's impartiality in the proceedings. See Christo, 223 F.3d at 1333. Accordingly, we decide that the bankruptcy judge did not abuse his discretion in denying Lawrence's motion for recusal.

III. CONCLUSION

Lawrence appeals the district court's affirmance of the bankruptcy court's orders determining that his obligations under a state court divorce decree are partially dischargeable and partially non-discharge-able, and denying his motion for recusal of the bankruptcy judge. Lawrence argues that (1) the bankruptcy court erred in finding that it was barred by the "law of the case" doctrine from independently re-determining what portion of the underlying state divorce court judgment was intended to be "in the nature of support," as opposed to a property settlement; and (2) the bankruptcy judge abused his discretion in denying Lawrence's motion for recusal. The bankruptcy court correctly held that it was bound by the "law of the case" doctrine from reexamining what portion of the divorce court's award was in the nature of support, and the bankruptcy judge did not abuse his discretion in denying Lawrence's motion for recusal because the two challenged comments were isolated, did not arise from an extrajudicial source, and would not lead a reasonable observer to question the judge's impartiality. The district court's order is AFFIRMED.