Synthesis of Kallilolide (J. A. Marshall, P. S. Coan 1995, , 796)
7) Dehydration of primary alcohol to alkene using a Lewis acid (aluminium trichloride in this case).
1. To introduce basic concepts of organic synthesis:
When there is a substituent on the central carbon of a phosphonate, a trisubstituted alkene will be formed on reaction with an aldehyde. The stereochemical preferences here are complicated, but diligent trial-and-error experiments have revealed that steric effects in the phosphonate can reverse the normal preference for the formation of -alkenes, to form predominantly -alkenes. In the examples below, the alkoxy substituents on both the ester and the phosphonate are progressively increased in size, resulting in a switch from predominantly to mostly stereochemistry (Nagaoka, H.; Kishi, Y. 1981, , 3873).

A pair of carbonyl compounds can also be reductively coupled together (with reduction) to generate an . Symmetrical alkenes can be prepared from a single aldehyde or ketone coupling with itself, using Ti metal reduction (the McMurry ). If two different ketones are to be coupled, a more complex, indirect method such as the Barton-Kellogg reaction may be used.
sulphuric acid to alkene now on separate page

A convergent total synthesis of (−)-nahuoic acid Ci(Bii) (3), a novel -decalin polyketide, has been achieved. Key synthetic transformations include Type II Anion Relay Chemistry (ARC) to construct the polyol chain, a Ti-catalyzed asymmetric Diels–Alder reaction to generate the -decalin skeleton, and a late-stage large fragment union exploiting a Micalizio alkoxide-directed alkyne–alkene coupling tactic.
Electrophilic addition reaction of bromine, electrophilic addition to alkenes with pure bromine or in non-polar solvent (non-aqueous Br2(l/solvent)) to give dibromoalkanes or electrophilic addition using bromine water [aqueous Br2(aq)] to give bromo-alcohols.
aq) and HBr(g/non-polar solvent)] to form halogenoalkanes

On the plus side, you've clearly thought about this problem and have worked out a detailed synthetic route. I think most of the steps would work as desired. The two steps that concern me the most are 4 and 7. I suspect your step 4 will give a mixture of products. Perhaps with further experimentation you could find reaction conditions that would give you the desired product without other products from undesired side reaction (elimination in the other direction and perhaps some rearrangement products too). I see your step 7 as the most problematical. Aluminum chloride, heat and the alcohol will produce a carbocation and it will undergo significant rearrangement creating a variety of elimination and addition (polymer formation) products. So while your reaction scheme is plausible, in my opinion it is not efficient.
A single ketone can also be converted to the corresponding alkene via its tosylhydrazone, using sodium methoxide (the Bamford-Stevens reaction) or an alkyllithium (the Shapiro reaction).

Alkynes and Alkenes Synthesis | Alkene | Unit Processes
-
Properties, Synthesis and Reactions of Alkenes and Alkynes
Presence of neighbouring carbonyl group increases the acidity of a ketone over an alkane by a factor of 1040!
-
Lab 4- Synthesis of Alkenes Flashcards | Quizlet
5)The enolate ions formed on deprotonation can be alkylated and acylated offering useful products.
-
Alkenes: Reactions and Synthesis Flashcards | Quizlet
Phanstiel O., Dueno E., and Wang Q. X., (1998), Synthesis of exotic soaps in the chemistry laboratory, , 75, 612.
Synthesis of terminal alkynes - Organic Chemistry Portal
Fig. 2. Fully controlled preparation of alkene isomers by eliminative cross-coupling of carbenoids. Understanding the concept at a heuristic level (A), representative results (B), and some significant biologically active targets that could benefit from application of the method (C).
Alkane or alkene synthesis by alkyne reduction
Proof-of-principle for eliminative cross-coupling was established by combining carbenoids containing lithium and oxygen with carbenoids containing boron and oxygen (Fig. 2B). As anticipated, like and unlike combinations of the pairs of carbenoids led to different geometrical isomers of the desired alkenes and with selectivity typically surpassing 19:1. The scope of the process as demonstrated was limited to simple trisubstituted alkenes of the styrene class and it remains to extend it to a wider variety of alkene targets and to improve on the chemical yield.
Alkene synthesis by isomerization - Organic chemistry
Olefination and alkyne elementometalation are two strategies commonly used to prepare alkenes but each method has major limitations. The first approach completely controls the type of regioisomer made but not which geometrical isomer is produced. By contrast, the second approach completely controls which geometrical isomer is generated but not which regioisomer. In cases where two or more groups Ri are very similar to one another it becomes difficult, or else impossible, to effectively make a single alkene isomer using these strategies (Fig. 1B & 1C).
Alkene Reactivity - Home - Michigan State University
In sum, eliminative cross-coupling makes it possible to precisely target a desired alkene isomer regardless of the nature of the structural differences responsible for distinguishing isomers. As such, this method has potential for the direct synthesis of biologically active alkenes possessing isomers that are all but impossible to access selectively with traditional approaches (Fig. 2C).
Alkene Reactivity - Michigan State University
Fig. 1. An alkene is a molecule containing a carbon-carbon double bond (C=C). Six distinct alkene isomers are possible if all Ri about C=C bond are different (A). Carbonyl compound olefination (B) and alkyne elementometalation (C) fail to give acceptable levels of selectivity for one alkene isomer over another when groups Ri are similar in character to each other.