AMENDMENT NO. 2 TO UNDERTAKING AGREEMENT

Exhibit 4.23(b)

AMENDMENT NO. 2 TO UNDERTAKING AGREEMENT

AMENDMENT dated as of October 27, 2004 to the Undertaking Agreement dated as of December 17, 2003 (as heretofore amended, the “Undertaking Agreement”) by LYONDELL CHEMICAL COMPANY, a Delaware corporation (“Lyondell”), in favor of the PURCHASERS (as defined in the Receivables Purchase Agreement dated as of December 17, 2003 (the “Receivables Agreement”) among Lyondell Funding II, LLC, as Seller, Lyondell, as Servicer, the Purchasers from time to time party thereto and Citicorp USA, Inc., as asset agent and administrative agent (the “Agent”) for the Purchasers) and CITICORP USA, INC., as Agent.

The parties hereto agree as follows:

SECTION 1. Defined Terms; References. Unless otherwise specifically defined herein, each term used herein which is defined in the Undertaking Agreement, or incorporated by reference therein, has the meaning assigned to such term in the Undertaking Agreement or so incorporated. Each reference to “hereof”, “hereunder”, “herein” and “hereby” and each other similar reference and each reference to “this Agreement” and each other similar reference contained in the Undertaking Agreement shall, after this Amendment becomes effective, refer to the Undertaking Agreement as amended hereby.

SECTION 2. Amendment to Article 3. Clause (e) of Section 3.16 of the Undertaking Agreement is amended to read in its entirety as follows:

(e) (i) cause one or more JV Subsidiaries to retain the material rights (or the right to reacquire the material rights) of Lyondell Petrochemical G.P. Inc. under the Equistar Partnership Agreement and (ii) maintain the rights of Lyondell Refining Company to appoint, together with CITGO Gulf Coast Refining, Inc., the chief executive officer of LCR, as provided in the LCR Partnership Agreement.

SECTION 4. Representations of Lyondell. Lyondell represents and warrants that (i) the representations and warranties of Lyondell set forth in Article 2 of the Undertaking Agreement and Article IV of the Receivables Agreement will be true in all material respects on and as of the Amendment Effective Date (as defined below) with the same effect as though made on and as of such date, except to the extent such representations and warranties expressly relate to an earlier date and (ii) no Potential Event of Termination or Event of Termination will have occurred and be continuing on such date.

SECTION 5. Governing Law. This Amendment shall be governed by and construed in accordance with the laws of the State of New York.

SECTION 6. Counterparts. This Amendment may be signed in any number of counterparts, each of which shall be an original, with the same effect as if the signatures thereto and hereto were upon the same instrument.

SECTION 7. Effectiveness. This Amendment shall become effective on the date (the “Amendment Effective Date”) when the Agent shall have received counterparts hereof signed by each of the Required Purchasers and Lyondell (or, in the case of any party as to which an executed counterpart shall not have been received, the Agent shall have received in form satisfactory to it facsimile or other written confirmation from such party of execution of a counterpart hereof by such party). Promptly after the Amendment Effective Date occurs, the Agent shall notify Lyondell and the Purchasers thereof, and such notice shall be conclusive and binding on all parties hereto.

2

IN WITNESS WHEREOF, the parties hereto have caused this Amendment to be duly executed as of the date first above written.