Exhibit 10.4

PRIVATE PLACEMENT SHARES PURCHASE AGREEMENT

THIS PRIVATE PLACEMENT SHARES PURCHASE AGREEMENT, dated as of September 25, 2025 (as it may from time to time be amended, this “Agreement”), is entered into by and between American Exceptionalism Acquisition Corp. A, a Cayman Islands exempted company (the “Company”), and AEXA Sponsor LLC, a Cayman Islands limited liability company (the “Purchaser”).

WHEREAS, the Company intends to consummate an initial public offering (the “Public Offering”) of the Company’s Class A ordinary shares, par value $0.0001 per share (each, a “Share”), as set forth in the Company’s Registration Statement on Form S-1 (File Number 333-289701), filed with the U.S. Securities and Exchange Commission (the “SEC”) under the Securities Act of 1933, as amended (the “Securities Act”); and

WHEREAS, the Purchaser has agreed to purchase an aggregate of 175,000 Shares (the “Private Placement Shares”).

NOW THEREFORE, in consideration of the mutual promises contained in this Agreement and other good and valuable consideration, the receipt and sufficiency of which are hereby acknowledged, the parties to this Agreement, intending legally to be bound, hereby agree as follows:

Section 1. Authorization, Purchase and Sale; Terms of the Private Placement Shares.

A. Authorization of the Private Placement Shares. The Company has duly authorized the issuance and sale of the Private Placement Shares to the Purchaser.

B. Purchase and Sale of the Private Placement Shares. On the date of the consummation of the Public Offering or on such earlier date as may be mutually agreed by the Purchaser and the Company (the “Closing Date”), the Company shall issue and sell to the Purchaser, and the Purchaser shall purchase from the Company, 175,000 Private Placement Shares at a price of $10.00 per Private Placement Share for an aggregate purchase price of $1,750,000 (the “Purchase Price”), which shall be paid by wire transfer of immediately available funds to the Company at least one (1) business day prior to the Closing Date in accordance with the Company’s wiring instructions. On the Closing Date, following the payment by the Purchaser of the Purchase Price by wire transfer of immediately available funds to the Company, the Company, at its option, shall deliver a certificate evidencing the Private Placement Shares purchased by the Purchaser on such date duly registered in the Purchaser’s name to the Purchaser or effect such delivery in book-entry form.

C. Terms of the Private Placement Shares.

(i) The Private Placement Shares are substantially identical to the Shares to be offered in the Public Offering except that (a) the Private Placement Shares will not, except in limited circumstances, be transferable, assignable or salable until 30 days after the Company has completed a merger, share exchange, asset acquisition, share repurchase, reorganization or

similar business combination with one or more businesses (the “Initial Business Combination”) (consistent with the terms of the letter agreement to be entered into by and among the Company, the Purchaser and the other parties thereto in connection with the Public Offering (the “Letter Agreement”)), and (b) the Private Placement Shares are being purchased pursuant to an exemption from the registration requirements of the Securities Act and will become freely tradable only after (i) the expiration of the lockup described above in clause (a) and (ii) they are registered pursuant to the Registration Rights Agreement (as defined below), or an exemption from registration is available.

(ii) Prior to or on the date of the consummation of the Public Offering, the Company and the Purchaser shall enter into a registration rights agreement (the “Registration Rights Agreement”) pursuant to which the Company will grant certain registration rights to the Purchaser relating to the Private Placement Shares.

Section 2. Representations and Warranties of the Company. As a material inducement to the Purchaser to enter into this Agreement and purchase the Private Placement Shares, the Company hereby represents and warrants to the Purchaser (which representations and warranties shall survive the Closing Date) that:

A. Organization and Corporate Power. The Company is an exempted company duly incorporated, validly existing and in good standing under the laws of the Cayman Islands and is qualified to do business in every jurisdiction in which the failure to so qualify would reasonably be expected to have a material adverse effect on the financial condition, operating results or assets of the Company. The Company possesses all requisite corporate power and authority necessary to carry out the transactions contemplated by this Agreement.

B. Authorization; No Breach.

(i) The execution, delivery and performance of this Agreement have been duly authorized by the Company as of the Closing Date. This Agreement constitutes the valid and binding obligation of the Company, enforceable in accordance with its terms, except that such enforcement may be subject to bankruptcy, insolvency, fraudulent conveyance, reorganization, moratorium and other laws of general applicability relating to or affecting creditors’ rights and to general equitable principles (whether considered in a proceeding in equity or law).

(ii) The execution and delivery by the Company of this Agreement, the issuance and sale of the Private Placement Shares and the fulfillment of and compliance with the terms hereof by the Company, do not and will not as of the Closing Date (a) conflict with or result in any violation or breach of the terms, conditions or provisions of, or constitute a default under, any law, statute, rule, regulation, agreement, order, judgment or decree to which the Company is subject, (b) result in or give rise to the creation of any lien, security interest, charge or encumbrance upon the Company’s share capital or assets, (c) conflict with or result in any violation of any provision of the amended and restated memorandum and articles of association of the Company (as in effect on the date hereof or as may be amended on or prior to the Closing Date), or (d) require any authorization, consent, approval, exemption or other action by or notice or declaration to, or filing with, any court or administrative or governmental body or agency, except for any filings required after the date hereof under federal or state securities laws.

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C. Title to Securities. Upon issuance in accordance with, and payment pursuant to, the terms hereof and upon registration in the register of members of the Company, the Private Placement Shares will be duly and validly issued, fully paid and nonassessable. Upon issuance in accordance with, and payment pursuant to, the terms hereof, the Purchaser will have good and valid title to the Private Placement Shares purchased by it, free and clear of all liens, claims and encumbrances of any kind, other than (i) transfer restrictions hereunder and under the other agreements contemplated hereby, including the Letter Agreement, (ii) transfer restrictions under federal and state securities laws, and (iii) liens, claims or encumbrances imposed due to the actions of the Purchaser.

D. Governmental Consents. No permit, consent, approval or authorization of, or declaration to or filing with, any governmental authority is required in connection with the execution, delivery and performance by the Company of this Agreement or the consummation by the Company of any other transactions contemplated hereby, except for any filings required after the date hereof under federal or state securities laws.

E. Regulation D Qualification. Neither the Company nor, to its knowledge, any of its affiliates, members, officers, directors or beneficial shareholders of 20% or more of its outstanding securities, has experienced a disqualifying event as enumerated pursuant to Rule 506(d) of Regulation D under the Securities Act.

Section 3. Representations and Warranties of the Purchaser. As a material inducement to the Company to enter into this Agreement and issue and sell the Private Placement Shares to the Purchaser, the Purchaser hereby represents and warrants to the Company (which representations and warranties shall survive the Closing Date) that:

A. Organization and Requisite Authority. The Purchaser possesses all requisite power and authority necessary to carry out the transactions contemplated by this Agreement.

B. Authorization; No Breach.

(i) This Agreement constitutes a valid and binding obligation of the Purchaser, enforceable in accordance with its terms, except that such enforcement may be subject to bankruptcy, insolvency, fraudulent conveyance, reorganization, moratorium and other laws of general applicability relating to or affecting creditors’ rights and to general equitable principles (whether considered in a proceeding in equity or law).

(ii) The execution and delivery by the Purchaser of this Agreement, and the fulfillment of and compliance with the terms hereof by the Purchaser, does not and shall not as of the Closing Date conflict with or result in a breach by the Purchaser of the terms, conditions or provisions of any agreement, instrument, order, judgment or decree to which the Purchaser is subject.

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C. Investment Representations.

(i) The Purchaser is acquiring the Private Placement Shares for its own account, for investment purposes only and not with a view towards, or for resale in connection with, any public sale or distribution thereof.

(ii) The Purchaser is an “accredited investor” as such term is defined in Rule 501(a)(3) of Regulation D under the Securities Act, and the Purchaser has not experienced a disqualifying event as enumerated pursuant to Rule 506(d) of Regulation D under the Securities Act.

(iii) The Purchaser understands that the Private Placement Shares are being offered and will be sold to it in reliance on specific exemptions from the registration requirements of the United States federal and state securities laws and that the Company is relying upon the truth and accuracy of, and the Purchaser’s compliance with, the representations and warranties of the Purchaser set forth herein in order to determine the availability of such exemptions and the eligibility of the Purchaser to acquire such Private Placement Shares.

(iv) The Purchaser did not decide to enter into this Agreement as a result of any general solicitation or general advertising within the meaning of Rule 502(c) of Regulation D under the Securities Act.

(v) The Purchaser has been furnished with all materials relating to the business, finances and operations of the Company and materials relating to the offer and sale of the Private Placement Shares which have been requested by the Purchaser. The Purchaser has been afforded the opportunity to ask questions of the executive officers and directors of the Company. The Purchaser understands that its investment in the Private Placement Shares involves a high degree of risk and it has sought such accounting, legal and tax advice as it has considered necessary to make an informed investment decision with respect to the acquisition of the Private Placement Shares.

(vi) The Purchaser understands that no United States federal or state agency or any other government or governmental agency has passed on or made any recommendation or endorsement of the Private Placement Shares or the fairness or suitability of the investment in the Private Placement Shares by the Purchaser nor have such authorities passed upon or endorsed the merits of the offering of the Private Placement Shares.

(vii) The Purchaser understands that: (a) the Private Placement Shares have not been and are not being registered under the Securities Act or any state securities laws, and may not be offered for sale, sold, assigned or transferred unless (1) in a registered transaction or (2) in reliance on an exemption therefrom and (b) except as specifically set forth in the Registration Rights Agreement, neither the Company nor any other person is under any obligation to register the Private Placement Shares under the Securities Act or any state securities laws or to comply with the terms and conditions of any exemption thereunder. While the Purchaser understands that Rule 144 under the Securities Act is not available for the resale of securities initially issued by shell companies (other than business combination related shell companies) or issuers that have been at any time previously a shell company, the Purchaser understands that Rule 144 includes an exception to this prohibition if the following conditions are met: (i) the issuer of the securities that was formerly a shell company has ceased to be a shell company; (ii) the issuer of

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the securities is subject to the reporting requirements of Section 13 or 15(d) of the Securities Exchange Act of 1934, as amended (the “Exchange Act”); (iii) the issuer of the securities has filed all Exchange Act reports and material required to be filed, as applicable, during the preceding 12 months (or such shorter period that the issuer was required to file such reports and materials), other than Form 8-K reports; and (iv) at least one year has elapsed from the time that the issuer filed current Form 10 type information with the SEC reflecting its status as an entity that is not a shell company.

(viii) The Purchaser has knowledge and experience in financial and business matters, knowledge of the high degree of risk associated with investments in the securities of companies in the development stage such as the Company, is capable of evaluating the merits and risks of an investment in the Private Placement Shares and is able to bear the economic risk of an investment in the Private Placement Shares in the amount contemplated hereunder for an indefinite period of time. The Purchaser has adequate means of providing for its current financial needs and contingencies and will have no current or anticipated future needs for liquidity which would be jeopardized by the investment in the Private Placement Shares. The Purchaser can afford a complete loss of its investment in the Private Placement Shares.

(ix) The Purchaser understands that the Private Placement Shares shall bear a legend and appropriate “stop transfer restrictions” substantially in the form set forth below:

“THE SECURITIES REPRESENTED HEREBY HAVE NOT BEEN REGISTERED UNDER THE UNITED STATES SECURITIES ACT OF 1933, AS AMENDED, OR ANY STATE SECURITIES LAWS, AND NEITHER THE SECURITIES NOR ANY INTEREST THEREIN MAY BE OFFERED, SOLD, TRANSFERRED, PLEDGED OR OTHERWISE DISPOSED OF EXCEPT PURSUANT TO AN EFFECTIVE REGISTRATION STATEMENT UNDER SUCH ACT OR SUCH LAWS OR AN EXEMPTION FROM REGISTRATION UNDER SUCH ACT AND SUCH LAWS WHICH, IN THE OPINION OF COUNSEL, IS AVAILABLE.

THE SECURITIES REPRESENTED BY THIS CERTIFICATE ARE SUBJECT TO LOCKUP PROVISIONS AND MAY NOT BE OFFERED, SOLD, TRANSFERRED, PLEDGED OR OTHERWISE DISPOSED OF DURING THE TERM OF THE LOCKUP.”

Section 4. Conditions of the Purchaser’s Obligations. The obligations of the Purchaser to purchase and pay for the Private Placement Shares are subject to the fulfillment (or to the extent permitted by applicable law, waiver by the Purchaser), on or before the Closing Date, of each of the following conditions:

A. Representations and Warranties. The representations and warranties of the Company contained in Section 2 shall be true and correct at and as of the Closing Date as though then made.

B. Performance. The Company shall have performed and complied with all agreements, obligations and conditions contained in this Agreement that are required to be performed or complied with by it on or before the Closing Date.

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C. No Injunction. No statute, rule, regulation, executive order, decree, ruling or injunction shall have been enacted, entered, promulgated or endorsed by or in any court or governmental authority of competent jurisdiction or any self-regulatory organization having authority over the matters contemplated hereby, which prohibits the consummation of any of the transactions contemplated by this Agreement.

D. Registration Rights Agreement. The Company shall have entered into the Registration Rights Agreement on terms satisfactory to the Purchaser.

Section 5. Conditions of the Company’s Obligations. The obligations of the Company to the Purchaser under this Agreement are subject to the fulfillment (or to the extent permitted by applicable law, waiver by the Company), on or before the Closing Date, of each of the following conditions:

A. Representations and Warranties. The representations and warranties of the Purchaser contained in Section 3 shall be true and correct at and as of the Closing Date as though then made.

B. Performance. The Purchaser shall have performed and complied with all agreements, obligations and conditions contained in this Agreement that are required to be performed or complied with by the Purchaser on or before the Closing Date.

D. Registration Rights Agreement. The Company shall have entered into the Registration Rights Agreement on terms satisfactory to the Company.

Section 6. Termination. This Agreement may be terminated at any time after December 31, 2025 upon the election by either the Company or the Purchaser upon written notice to the other party if the Closing Date does not occur prior to such date.

Section 7. Survival of Representations and Warranties. All of the representations and warranties contained herein shall survive the Closing Date.

Section 8. Miscellaneous.

A. Successors and Assigns. Except as otherwise expressly provided herein, all covenants and agreements contained in this Agreement by or on behalf of any of the parties hereto shall bind and inure to the benefit of the respective successors of the parties hereto whether so expressed or not. Notwithstanding the foregoing or anything to the contrary herein, no party hereto may assign this Agreement without the prior written consent of the other party, other than assignments by the Purchaser to affiliates thereof (including, without limitation, one or more of its members).

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B. Severability. Whenever possible, each provision of this Agreement shall be interpreted in such manner as to be effective and valid under applicable law, but if any provision of this Agreement is held to be prohibited by or invalid under applicable law, such provision shall be ineffective only to the extent of such prohibition or invalidity, without invalidating the remainder of this Agreement.

C. Counterparts. This Agreement may be executed simultaneously in two or more counterparts, none of which need contain the signatures of more than one party, but all such counterparts taken together shall constitute one and the same agreement. Signatures to this Agreement transmitted via electronic transmission shall be valid and effective to bind the party so signing.

D. Descriptive Headings; Interpretation. The descriptive headings of this Agreement are inserted for convenience only and do not constitute a substantive part of this Agreement. The use of the word “including” in this Agreement shall be by way of example rather than by limitation.

E. Governing Law. This Agreement shall be deemed to be a contract made under the laws of the State of New York and for all purposes shall be construed in accordance with the internal laws of the State of New York, without giving effect to conflicts of law principles that would result in the application of the laws of another jurisdiction.

F. Amendments. This Agreement may not be amended, modified or waived as to any particular provision, except by a written instrument executed by all parties hereto.

[Signature page follows]

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IN WITNESS WHEREOF, the parties hereto have executed this Agreement to be effective as of the date first set forth above.


COMPANY:

AMERICAN EXCEPTIONALISM ACQUISITION CORP. A

By:		/s/ Jeffrey Vignos
Name:		Jeffrey Vignos
Title:		Chief Financial Officer

PURCHASER:

AEXA SPONSOR LLC

By:		/s/ Jeffrey Vignos
Name:		Jeffrey Vignos
Title:		Chief Financial Officer

American Exceptionalism Acquisition Corp. A 8-K