The answer to this, I recon, *is that they don't*.

They use molecular oxygen (O<sub>2</sub>) dissolved in the water for respiration, where is acts as a terminal electron acceptor, just as we use molecular oxygen in the air for respiration. We can speak of the water as being *oxygenated*. 

What *is* split in photosynthesis, where **reducing equivalents from water** are used to reduce NADP<sup>+</sup> (giving NADPH).

One of the great discoveries of biology, IMO, **is that the oxygen formed in green-plant photosynthesis comes from water, not CO<sub>2</sub>**.


Despite claims to the contrary, most infamously by Racker (1976, pp 28 - 29), but also by (for example) Madeira (1988), water is *not* split in the tricarboxylic acid cycle (Krebs Cycle). That is, reducing equivalents from water are *not* passed down the respiratory chain, or in any way used to make ATP, or are in any way a 'source' of free energy.  Such claims, IMO, are nonsense.

Great question, BTW.

**Edit**

In order to avoid opening a can of worms, I decided to dig out the old references on the splitting of water in the TCA cycle controversy. The claim that water is split in the TCA cycle is also made by Wieser (1980), and is repeated by Mego (1986).  Banfalvi (1991) also sails pretty close to the wind.  

The definitive answers to the Weiser (1980) paper are given by Atkinson (1981) and Herreros & Garcia-Sancho (1981). Both of these articles are models of clarity, and categorically refute the claims of Weiser (1980).  Nevertheless, as shown by the references above, the controversy surfaces periodically.

I'll give some quotes from each of these papers, which are also directly relevant the OP question.

> - Ionization of water neither yields nor consumes electrons, and could not possible provide electrons for anything.    

>- In photosynthesis,
> water loses electrons, **being oxidized to O<sub>2</sub>**; in respiration
> O<sub>2</sub> gains electrons, **being reduced to water**. Textbooks
> are not in error in making these points, and of course the roles of
> the water-oxygen system in photosynthesis and in respiration are
> oppositely directed, not similar as Weiser says.  

>[from Atkinson (1981). The emphasis is mine]

And

> - ...the carbon compounds themselves are the only source of reducing power in the TCA cycle
> - It is true that one molecule of glucose can not supply 12 pairs of H, but it can certainly supply 12 pairs of H.
> - **In the TCA cycle, the electrons shared in the C-C and C-H bonds are the source of reducing power; they are transferred first to pyridine nucleotides (and flavin nucleotide) and then to the vicinity of oxygen through the respiratory chain**

>[From Herreros & Garcia-Sancho (1981).  The emphasis is mine]

As you may have guessed, the splitting of the water in the TCA cycle is a pet rant of mine.  Thanks for the opportunity of airing my views! 

**References**

(Apologies for incomplete refs.  I'll update these later on)

 - Atkinson, D.E. (1981) TCA Cycle Confusion.  *Trends in Biochemical Sciences* (full ref to follow)

 - Banfalvi,G. (1991)  Conversion of Oxidative Energy to Reductive Power in the Citrate Cycle. *Biochemical Education*, **19**, 24 - 26  [see [here](http://onlinelibrary.wiley.com/doi/10.1016/0307-4412%2891%2990138-X/abstract)] (pdf apparently free to all)


 - Herreros, B. & Garcia-Sancho, J. (1981) TCA Cycle Confusion.  *Trends in Biochemical Sciences* (full ref to follow)

 - Madeira, V.M.C. (1988) Stoichiometry of Reducing Equivalents and Splitting
of Water in the Citric Acid Cycle.  *Biochemical Education* **16**, 94 - 96 [[pdf](http://onlinelibrary.wiley.com/doi/10.1016/0307-4412%2888%2990069-6/pdf)] (apparently free to all.)

 - Mego, J.L. (1986)  The Role of Water in Glycolysis *Biochemical Education*,  **14**, 130 - 131. (see [here](http://onlinelibrary.wiley.com/doi/10.1016/0307-4412%2886%2990175-5/abstract))
 - Racker, E. (1976) A New Look at Mechanisms in Bioenergetics.  Academic Press, New York.
 - Weiser (1980) Textbook Errrors: The splitting of water by the tricarboxylic acid cycle.  Textbook error or textbook omission? *TIBS* (1980) **5** (Issue 11), 284. [see [here](http://www.sciencedirect.com/science/journal/09680004/5/11)].  [[pdf](http://download.cell.com/trends/biochemical-sciences/pdf/PII0968000480901590.pdf?intermediate=true)], apparently free to all.