Lindsay's own biographical summary (see the biography page) mentions that his older brother wrote under the pseudonym Alexander Crawford, and names one novel, The Alias. He adds that his brother's novels "now are almost forgotten".
The British Library integrated catalogue lists 4 titles for Crawford, Alexander (novelist): Kapak (1911), Monsieur Carnifex (1912), The Alias (1913) and Outside the Law (1914), all published by William Blackwood & Sons.
A page at the Library of the University of North Carolina website lists the content of a collection of documents from the early days of the literary agent A P Watt, among which are contracts for the abovementioned works, as well as several other titles by Alexander Crawford, which may be novels or stories. The additional titles are: An Uncharted Reef, Undue Influence, a translation of Mr Moneypenny, The Experiment, The Husband She Bought, La Petrelle, Mr Moysey's First Case, Just Like Other Men, A Scrap of Paper, as well as an unnamed serial for The Evening News.
Of Lindsay's relation with his older brother, J B Pick says:
[David] was never close to his brother, who died comparatively young and was a hard-living journalist, much entangled with women. Some of his adventures had repercussions on the family that deeply disturbed David.
Bernard Sellin (The Life & Works of David Lindsay, p.15) says that "Alexander's hectic life ended abruptly at the age of forty, when he died from a liver ailment". Alexander Lindsay's death certificate gives his age as 46.
Here is a review from The Times, 16th March 1914, for one of Crawford's novels:
More information about Alexander Crawford can be found in Douglas A Anderson's article, "Resurrecting Alexander Crawford" in Wormwood #11 (Autumn 2008), which also reprints a story by Crawford, "The Experiment". Wormwood is available from Tartarus Press.